REVIEW. Final Exam. Final Exam Information. Final Exam Information. Strategy for Studying. Test taking strategy. Sign Convention Rules

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1 Final Exam Information REVIEW Final Exam (Print notes) DATE: WEDNESDAY, MAY 12 TIME: 1:30 PM - 3:30 PM ROOM ASSIGNMENT: Toomey Hall Room Final Exam Information Comprehensive exam covers all topics on syllabus. Equivalent to taking two regular exams. Format is similar to what you have seen on hourly exams throughout the semester. Strategy for Studying Review topics from the first half of the semester. WORK problems on topics that you are most uncomfortable with first. Work problems on topics that you are confident with last. 3 4 Test taking strategy Get to bed by midnight the night before. Eat a good breakfast. Work the problems you CAN and go back to the rest later. If you have time, take a two minute mental break and check over the whole exam once before turning in. Make sure you show all work Make sure your work is orderly 5 Sign Convention Rules Always use a right handed coordinate system Designate the equation you are writing and indicate positive direction. Use right hand rule when evaluating moments. 6 Review_Final Exam 1

2 Types of vectors: Position vectors ( r ) Unit vectors ( u ) Vectors Vectors expressing a force ( F ) / moment ( M ) Vector operations: Addition / subtraction Dot product i*i + j*j + k*k Scalar answer (NO < i, j, k > components; just magnitude) Particle Equilibrium - 2D & 3D Static Equilibrium Equations: ΣF X ΣF Y (for 2D) ΣF Z (for 3D) Solving these problems: Draw Free Body Diagram of point! Springs add additional equation F SP = kδs Be able to do 3x3 matrix on calculator ΣF = 0 (for equilibrium problems) OR ΣF = F R (for resultant force) 3D: Often better to use vector notation 7 8 Moment Created by Forces Moment Created by Forces Static Equilibrium Equation: Solving these problems: Draw Free Body Diagram of point! Scalar notation (M = F*d): Make sure moment arm is to force ΣM POINT Solving these problems ( continued) : Vector notation ( M = r x F ): Watch direction of position vector ( r ) r is always first in cross product. Moment about a line: Watch sign convention and direction of moment created by each component of each force. Find moment about point ON line Then M = M u line 9 10 Couple moments Can sum moment for two forces creating a couple at any point and the resulting moment will be the same value. Equivalent Systems Locate point of interest. If you have to find it, then use the problem to help you find a preliminary stand in point. Sum forces (ΣF = F R ) Sum moments at point of interest (ΣM = M R ) If problem asks for resultant force ONLY, then find location (d) using equation M R = F R *d Review_Final Exam 2

3 Distributed Loads Reduce to a single force by integrating the area under the load diagram. Find location of resultant force using the same method as for equivalent 13 Equilibrium of a 2D Rigid Body Equilibrium Equations for 2D: ΣF X ΣF Y ΣM point Solving these problems: Identify all forces and support reactions Draw Free Body Diagram of object! Sum forces in x and y direction. Sum moments at a point Don t forget couple moments! 14 Trusses: 2D Rigid Bodies Consist entirely of 2-force members. Direction of forces in members act along the direction of member (axially) Note axial direction of forces using (T) for tension and (C) for compression. DROP NEG. SIGN!! Analyze forces by cutting through internal members 15 Analyzing Trusses When cutting through members, this results in Free Body Diagrams of: Single joint / pin connection Max of 2 unknowns ΣFX ΣFY Section of structure (two or more joints) Max of 3 unknowns ΣFX ΣFY ΣMpoint Always assume unknown forces to be in tension (positive) negative answer indicates compression 16 Frames and Machines DO NOT consist entirely of 2-force members. Know how to identify 2-force members to help eliminate unknowns DO NOT cut through members, but take the structure apart at the connections. Direction of forces acting at connections is unknown, so make an assumption. Make sure forces on contacting members act in opposite directions. 17 Analyzing Frames & Machines Write 3 equilibrium equations for each Free Body Diagram. You may have more than 3 unknowns on a particular diagram, so just solve what you can and move on Using the process of elimination to solve unknowns. The TOTAL number of unknowns cannot exceed the TOTAL number of equilibrium equations you can write. 18 Review_Final Exam 3

Comparison Table Internal Structural Loads Characteristics Consist of members FBDs Assume direction of unknown is Trusses 2-force Cut through members always tension Frames & Machines Multiforce Take

internal loads so they become external 19 20 Sign Convention for Internal Loads Types of Problems: Evaluating Internal Loads 1. Cut through structure at specific point. 2. Cut through structure at arbitrary point.

4 Comparison Table Internal Structural Loads Characteristics Consist of members FBDs Assume direction of unknown is Trusses 2-force Cut through members always tension Frames & Machines Multiforce Take apart at joints Doesn t matter (except for cables) Multi-force members carry Moment load: bending (M) or torsional (T) Shear force (V) Normal force (N) To evaluate, cut through members Expose internal loads so they become external Sign Convention for Internal Loads Types of Problems: Evaluating Internal Loads 1. Cut through structure at specific point. 2. Cut through structure at arbitrary point. 3. Integrate load on beam (2x) and solve for constants of integration (p ). 4. Construct diagrams using area method. (does not give equations, just a graph) Arbitrary Cut Whenever there is a change in load, a discontinuity is created. Divide beam into regions determined by discontinuity. The beam shown below has three distinct regions bounded by (0 x < 3) (3 x < 5) (5 x < 8) Set origin (x = 0) at left end of beam. Cannot evaluate internal shear EXACTLY at the point of an applied force. Must be just to the right or left of load. Cut through structure at arbitrary point. 23 Plot directly under beam x = 0 at left end Plot shear first Then plot moment Shear and moment are measured on y-axis Label each axis and graph including units Label all minimums and maximums, including points where shear crosses x-axis. SHOW calculations used to get values and draw graphs. Drawing V & M Diagrams 24 Review_Final Exam 4

Six Rules for V & M Diagrams 1. Concentrated force creates a jump in the shear diagram Creates downward jump Creates upward jump 2. Change in shear equals area under load diagram 3.

Concentrated moment creates a jump in moment diagram Creates downward jump Creates upward jump 25 x = P h Dry Friction (F s = µ s N) We always evaluate the case for impending motion # of unknowns = #

Position of applied force influences how object responds 26 Friction Wedges Uses the same friction equation as for dry friction there are just more pieces.

5 Six Rules for V & M Diagrams 1. Concentrated force creates a jump in the shear diagram Creates downward jump Creates upward jump 2. Change in shear equals area under load diagram 3. Slope of shear diagram equals value of distributed load 4. Change in moment equals area under shear diagram 5. Slope of moment diagram equals value of shear diagram. 6. Concentrated moment creates a jump in moment diagram Creates downward jump Creates upward jump 25 x = P h Dry Friction (F s = µ s N) We always evaluate the case for impending motion # of unknowns = # of equil. eqns. + friction eqn. Dimensions must be given to check for tipping. ΣM o to check for tipping If (x < l / 2) then block will slide. If (x > l / 2) then block will tip. Position of applied force influences how object responds 26 Friction Wedges Uses the same friction equation as for dry friction there are just more pieces. Be very careful when evaluating that you look at the direction of impending motion for each piece. Free Body Diagrams are absolutely essential. impending motion impending motion remains stationary 27 Belt Friction T 2 = T 1 e µβ T 2 is in the direction of impending motion T 1 opposes direction of impending motion µ = coefficient of friction between belt and surface of contact β = angle of contact between belt and surface 28 Finding Centroids Calculate as a weighted average: 1. Compute the moment of each integral element [weight, mass, volume, area, length] about an axis 2. Divide by total [weight, mass, volume, area, length] Using Single Integration 1) DRAW an element on the graph. 2) Label the centroid (x, ~ ~ y) 3) Label the point where the element intersects the curve (x, y) 4) Write down the general equation 5) Define each term according the problem statement 6) Determine limits of integration 7) Integrate Review_Final Exam 5

Finding Centroids of Composite Shapes 1) Divide the object into simple shapes.

simple shape 4) Set up a table as shown below to calculate values 5) Subtract empty areas instead

Resultant Force for Fluid on Surface 31 Flat surfaces Solve for perpendicular resultant force _ F

6 Finding Centroids of Composite Shapes 1) Divide the object into simple shapes. 2) Establish a coordinate axis system on the sketch ~ ~ 3) Label the centroid (x, y) of each simple shape 4) Set up a table as shown below to calculate values 5) Subtract empty areas instead of adding them. Resultant Force for Fluid on Surface 31 Flat surfaces Solve for perpendicular resultant force _ F perp = γ z A or F perp = (w 1 +w 2 )(1/2)(L) Curved surfaces Solve for vertical and horizontal components of resultant force and THEN find resultant. F v = γ (volume) F h = F perp 32 Review_Final Exam 6

F R. + F 3x. + F 2y. = (F 1x. j + F 3x. i + F 2y. i F 3y. i + F 1y. j F 2x. ) i + (F 1y. ) j. F 2x. F 3y. = (F ) i + (F ) j. ) j

F R. + F 3x. + F 2y. = (F 1x. j + F 3x. i + F 2y. i F 3y. i + F 1y. j F 2x. ) i + (F 1y. ) j. F 2x. F 3y. = (F ) i + (F ) j. ) j General comments: closed book and notes but optional one page crib sheet allowed. STUDY: old exams, homework and power point lectures! Key: make sure you can solve your homework problems and exam problems.