Tutorial 02- Statics (Chapter )

Transcription

1 Tutorial - Statics (Chapter ) MECH Spring 9 eb-6 th Tutor: LEO Contents Reviews ree Bod Diagram (BD) Truss structure Practices

2 Review -orce vector, Moment, Equilibrium of orces, D/3D Vectors (position, force) Adding/Subtracting vectors (Parallelogram law/trigonal) Resolution of a vector using Cartesian vector notation (CVN) Addition using CVN Multiplication of vectors (dot product) Moment / torque (D/3D) Cross product M O r Tpical supports Review -orce vector, Moment, Equilibrium of orces The equivalent sstem Moving a force from point A to O requires creating an additional couple moment. Since this new couple moment is a free vector, it can be applied at an point P on the bod. When several forces and couple moments act on a bod, ou can move each force and its associated couple moment to a common point O. Afterwards, ou can add all the forces and couple moments together and find one resultant force-couple couple moment pair.

3 Review -- GROUP PROBLEM SOLVING Given: Handle forces and are applied to the electric drill. ind: An equivalent resultant force and couple moment at point O. Plan: a) ind RO Σ i Where, b) ind M RO Σ M C + Σ ( r i i ) i are the individual forces in Cartesian vector notation (CVN). M C are an free couple moments in CVN (none in this eample). R i are the position vectors from the point O to an point on the line of action of i. SOLUTION {6 i 3 j k} N { i + j 4 k} } N RO {6 i j 4 k} N r {.5 i +3.3 k} m r {-.5 j +.3 k} m M RO r + r M RO { i j k i j k } N m { {.9 i j.45 k +4.4 i +j j +k}n m {.3 i j.45 k} N m

4 Rigid Bod Equilibrium THE PROCESS O SOLVING RIGID BODY EQUILIBRIUM PROBLEMS or analzing an actual phsical sstem, first we need to create an idealized model. Then we need to draw a free-bod diagram showing all the eternal (active and reactive) forces. inall, we need to appl the equations of equilibrium to solve for an unknowns. Rigid bod: The distance between an two points within the bod is constant. M O Where point O is an arbitrar point. Note: Newton s 3rd LAW Rigid Bod Equilibrium IMPORTANT NOTES. If we have more unknowns than the number of findependent d equations, then we have a staticall indeterminate situation. We cannot solve these problems using just statics. ti. The order in which we appl equations ma affect the simplicit of the solution. or eample, if we have two unknown vertical forces and one unknown horizontal force, then solving X O first allows us to find the horizontal unknown quickl. 3. If the answer for an unknown comes out as negative number, then the sense (direction) of the unknown force is opposite to that assumed when starting the problem. 4. Two force members; Three force members. O 3 4

5 Rigid Bod Equilibrium ree Bod Diagram BD PROCEDURE OR DRAWING A REE BODY DIAGRAM. Draw an outlined shape. Imagine the bod to be isolated or cut free from its constraints and draw its outlined shape.. Show all the eternal forces and couple moments. These tpicall include: a) applied loads, b) support reactions, and, c) the weight of the bod (if applicable). 3. Label loads and dimensions: All known forces and couple moments should be labeled with their magnitudes and directions. or the unknown forces and couple moments, use letters like A, A, M A, etc.. Indicate an necessar dimensions. orce direction in free bod diagram Whether a force positive or negative depends on our own dfiiti definition. A B P A B P A A P B B P A B A P B P

6 Moment direction in free bod diagram M M O + M M + M M M + M M + M 3 3 O 3 3 M M O M + M M M M M + M M A General Case Aim: ind the reaction force on point A and B. A A B Equations of Equilibrium X Y M ( ) B A qa+ A B 5 qa a aa M a A A M 5 ( + qa) a M, B (3 + qa) a

7 orce Analsis in a Truss Structure Zero force Members: In Engineering mechanics, a zero force member refers to a member in a truss which, given a specific load, is at rest neither in tension nor in compression. In a truss a zero force member is often found at pins (connections within the truss) where no eternal load is applied and three or fewer truss members meet. How to identif a zero force member? Two principles: ) The geometr of a truss structure is fied. a) Rigid bod b) Triangles (Wikipedia: zero-force member) ) Inspect those potential two force members first, then the others. orce Analsis in a Truss Structure Zero-force Members in a Truss Structure Two principles: ) The geometr of a truss structure is fied. a) Rigid bod b) Triangles ) Inspect those potential two force members first, then the others. Eample: Indicate all zero-force members. Answer: AB,BC,CD,DE,HI, and GI

8 Joint Method + G E + A A A m B m C m D D kn Simplified the structure to a frame to solve out the support reactions. orce balance in X-direction: A orce balance in Y-direction: A + D - Moment balance at C: A kn, D66.67 kn A -D G E A33.33 D66.67 Joint A: A A Definition: The forces are applied to the joints b the beams. Joint B: m B m C m kn AB AG cos 45 AG sin kN, 47.kN AB (T) AG (C) D BG, BC 33.3 kn (T)

9 G E Joint D: Joint E: A33.33 A B C A m m m DC DE DE DC kn66.7 kn 94.9 kn94.3 kn DE kn + cos sin 45 (T) (C) EG 94.9sin cos 45 EC kn66.7 kn (T) EC kn66.7 kn (C) EG D66.67 D G E A33.33 D66.67 Joint C: A B C D A m m m kn 67 cos 45 + CG kn (T) CG

10 Joint Method. Joint Method + + M A + 3 R D cos3 3 R D 3.79kN A cos3 A 33.34kN A 3.79sin 3 A 5.4kN

11 Joint Method kn kn 3.79 kn Joint A: AB AB 33kN (C) A kN (T) A Joint Method kn kn 3.79 kn Joint B: 33 B 3.33kN3.3kN (T) B 33 BC BC 3.33kN3.3kN (C)

12 + + Joint : C C C 4.74kN4.7kN (C) E kN.3kN (T) E Joint E: 3kN (T) EC.6kN.3kN (T) ED + + Joint C: CD 37.7kN 37.7kN (C) CD (check)

13 Joint Method The Approach of Joint Method: ) BD ree bod diagram. Remove the boundaries; add back support reactions. ) Use EoE equilibrium equations (forces & moments) to solve out the support reactions. 3) Inspect the force condition in each joints. igure out the forces. 4) Indicate the members are in tension or compression. Section Method.

14 + Y + X Support Reactions: M E A E 7.333kN Moment balance at C M C G 3 sin 6 G 8.8kN (T) Moment balance at M CD 3 sin 6 CD 8.47kN (C) orce balance along Y direction C sin C.77kN (T) Stress condition.

15 + Y + X orce balance along Y direction 3 4 D kn (C) Moment balance at D 4 3 M G 3 D kN G (C) Moment balance at 4 M 5 6 DE kN (T) DE D Section Method Notes when using section method:. Imaginar cutting shall through the cross section of the member ou want to analze.. The cutoff rule is not necessaril straight. Th ti t b ti t lit th t t i t 3. The section must be continuous to split the structure into TWO parts.