ARCH 331 Note Set 3.1 Su2016abn. Forces and Vectors

Transcription

1 orces and Vectors Notation: = name for force vectors, as is A, B, C, T and P = force component in the direction = force component in the direction R = name for resultant vectors R = resultant component in the direction R = resultant component in the direction tail tip = start of a vector (without arrowhead) = direction end of a vector (with arrowhead) = ais direction = ais direction = angle, in a trig equation, e. sin, that is measured between the ais and tail of a vector Characteristics orces have a point of application tail of vector size units of lb, K, N, kn direction to a reference sstem, sense indicated b an arrow Classifications include: Static & Dnamic Structural tpes separated primaril into Dead Load and Live Load with further identification as wind, earthquake (seismic), impact, etc. Rigid Bod Ideal material that doesn t deform orces on rigid bodies can be internal within or at connections or eternal applied Rigid bodies can translate (move in a straight line) or rotate (change angle) Weight of truck is eternal (gravit) Push b driver is eternal Reaction of the ground on wheels is eternal If the truck moves forward: it translates If the truck gets put up on a jack: it rotates 63

2 Transmissibilit: We can replace a force at a point on a bod b that force on another point on the bod along the line of action of the force. Eternal conditions haven t changed or the truck: = The same eternal forces will result in the same conditions for motion Transmissibilit applies to EXTERNAL forces. INTERNAL forces respond differentl when an eternal force is moved. DEINITION: 2D Structure - A structure that is flat and ma contain a plane of smmetr. All forces on this structure are in the same plane as the structure. Internal and Eternal Internal forces occur within a member or between bodies within a sstem Eternal forces represent the action of other bodies or gravit on the rigid bod 64

3 orce Sstem Tpes Collinear all forces along the same line Coplanar all forces in the same plane Space not concurrent or coplanar (all out there in 3 dimensions) urther classification as Concurrent all forces go through the same point Parallel all forces are parallel Graphical Addition R Parallelogram law: when adding two vectors acting at a point, the result is the diagonal of the parallelogram The tip-to-tail method is another graphical wa to add vectors. R P P With 3 (three) or more vectors, successive application of the parallelogram law will find the resultant OR drawing all the vectors tip-to-tail in an order will find the resultant. Rectangular orce Components and Addition It is convenient to resolve forces into perpendicular components (at 90). Parallelogram law results in a rectangle. Triangle rule results in a right triangle. 65

4 is: between & = = cos sin magnitudes are scalar and can be negative & are vectors in and direction = 2 2 tan = When 90 < < 270, is negative When 180 < < 360, is negative When 0 < < 90 and 180 < < 270, tan is positive When 90 < < 180 and 270 < < 360, tan is negative Addition (analticall) can be done b adding all the components for a resultant component and adding all the components for a resultant component. R, 2 2 R and R R R tan R R CAUTION: An interior angle,, between a vector and either coordinate ais can be used in the trig functions. BUT No sign will be provided b the trig function, which means ou must give a sign and determine if the component is in the or direction. or eample, sin opposite side, which would be negative in! 66

5 Eample 1 (pg 26) Steps: 1. GIVEN: Write down what s given (drawing and numbers). 2. IND: Write down what ou need to find. (resultant graphicall) 3. SOLUTION: 4. Draw the 200 lb and 300 lb forces to scale with tails at O. (If the scale isn t given, ou must choose one that fits on our paper, ie. 1 inch =100 lb.) 5. Draw parallel reference lines at the ends of the vectors. 6. Draw a line from O to the intersection of the reference lines 7. Measure the length of the line 8. Convert the line length b the scale into pounds (b multipling b the force measure and dividing b the scale value, ie X inches * 100 lb / 1 inch). 9. Measure the angle with respect to the positive ais. 67

6 Alternate solution: 4. Draw one vector 5. Draw the other vector at the TIP of the first one (awa from the tip). 6. Draw a line from 0 to the tip of the final vector and continue at step 7 Eample mm = 1 lb. or 1mm = 2/3 lb. 68

7 Eample 3 (pg 30) 69

8 Eample 4 (pg 26) Determine the resultant vector analticall with the component method. This is the same problem as Eample Problems 2.3 which was solved earlier using the graphical methods. 70