ARC241 Structural Analysis I Lecture 1, Sections ST1.1 ST2.4

Transcription

1 Lecture 1, Sections ST1.1 ST2.4 ST1.1-ST1.2) Introduction ST1.3) Units of Measurements ST1.4) The International System (SI) of Units ST1.5) Numerical Calculations ST1.6) General Procedure of Analysis ST2.1) Scalars and Vectors ST2.2) Vector Operations ST2.3) Vector Addition of Forces ST2.4) Addition of System of Coplanar Forces

2 ST1.1) Mechanics Mechanics studies the behavior of bodies subjected to the action of forces, and can be divided into two parts: 1. Statics: Study of objects in equilibrium (at rest or moving with a constant velocity). 2. Dynamics: Study of objects with accelerated motion. Mechanics calculations are carried out using Scalar (signed magnitude only) and Vector (magnitude, direction, and sense) quantities. 1

3 ST1.2) Fundamental Quantities in Mechanics 1. Length, d: locates the position of a point in space (meters, m) Scalar. 2. Time, t: measures the period during which an action, process, or condition exists or continues (seconds, s) Scalar. 3. Mass, m: the quantity of matter that an object has (gram, g) Scalar Note that unlike weight, mass does not change from one location to another. 4. Force, F: push or pull exerted by one object o another (Newton, N) Vector. 2

4 The four basic quantities of mechanics are related by Newton s Second Law: F = ma Where a is the acceleration represented as: a = d t 2 Thus, F = md 2 t 3

5 Applying Newton s Second Law to calculate the earth gravitational force (Weight, W) by setting F to W, and a to g (gravitational acceleration), one gets: Where the earth gravitational acceleration at sea level is m approximately s 2 W = mg 4

6 ST1.4) SI Units SI, International System of Units (Système International d'unités) adopts the following prefixes for units: Number Exponential form Prefix SI Symbol tera T giga G mega M kilo k milli m micro µ nano n Note that SI standards only recognize exponent multiples of 3 and -3. This is also called engineering format. 5

7 Significant figures determines the precision of a number. The number of significant figures is the number of digits in a number, including zero. Example: Express 4300 and in 3 and 2 significant figures, respectively, using Engineering format = 4.30(10 3 ) = 240(10 6 ) 6

8 When reducing a number to n number of significant figures, it must be rounded off according to the following rules: 1. If the n +1 digit is less than 5, then the n+1digit and others following it are dropped (Ex and reduced to 2 significant figures are 2.3 and 0.45). 2. If the n+1 digit is equal to 5 with zeros following it, then round off the nth digit to an even number (Ex (10 3 ) and rounded to n=3 significant figures become 1.24(10 3 ) and If the n+1 digit is greater than or equal to 5 with any nonzero digits following it, then increase the nth digit by 1 and drop the n+1 digit and others following it. (Ex and rounded off to n = 3 become and

9 As it was indicated, a force vector has a magnitude, direction, and sense. Consider the following force vector: This force vector has a magnitude of 4 N, and a direction of 20 o counterclockwise from the horizontal axis. The sense is indicated by the arrow tip. 8

10 Multiplication of a vector A by a scalar a produces a vector of Magnitude a A and the same direction of A if a is positive and opposite direction of A if a is negative. 9

11 Two vectors A and B can be added to form a resultant vector R = A + B By the parallelogram law. They can also be added using triangle construction. 10

12 The subtraction of two vectors can be expressed as R` = A B = A + (-B). 11

13 A vector can be resolved into two components along known lines by applying parallelogram rule in reverse. 12

14 Equilibrium of Concurrent Forces There are two types of problems that deals with equilibrium of concurrent forces: 1. Finding the resultant force, knowing its components. 2. Resolving a known force into two components. Both of these types of problems can be solved using parallelogram law or triangle construction. If more than two forces are present, then problem can be solved by reapplying the methods on the resultant of each two forces with another force (but using rectangular-component method is easier). 13

15 Remember cos(a + b) = cos a cos b + sin a sin b sin (a + b) = sin a cos b + sin b cos a d(sinx) = cosx d(cosx) = -sinx 14

16 Problem ST2.3: Determine F R = F 1 + F 2 15

17 Problem ST2.8: Find θ and F R so that F R is directed horizontally to the right. 16

18 Problem ST2.25: Find theta, F A and F B so that F R is directed horizontally to the right with a magnitude of 950N, and F B is a minimum. 17

19 When the resultant of more than two forces has to be obtained, it is easier to: 1) Establish a coordinate system. 2) Find the components of each force along specified axes. 3) Add these components algebraically. 4) Form the resultant. F = F i x + F y j and F = Ry Fy 18

20 Problem ST2.34: Find F R 19