hwhat is mechanics? hscalars and vectors hforces are vectors htransmissibility of forces hresolution of colinear forces hmoments and couples

Transcription

1 orces and Moments CIEG-125 Introduction to Civil Engineering all 2005 Lecture 3 Outline hwhat is mechanics? hscalars and vectors horces are vectors htransmissibilit of forces hresolution of colinear forces hmoments and couples What is Mechanics? Rigid Bod Mechanics (also fluid mechanics, soil mechanics etc) - forces acting on bodies Statics - bodies at rest or moving with uniform velocit Dnamics - bodies accelerating Strength of materials - deformation of bodies under forces Structural Mechanics - focus on behavior of structures under loads Rigid bod is a bod that ideall does not deform under a force ll material deforms When deformations are small assume the bod is rigid. Eamples foam block with a coin wood block with a small weight stone arches Elasticall Deformable Bod Inelasticall Deformable Bod Bodies that undergo reversible deformations Eamples: rubber bands springs steel, concrete and wood structures under small deformations If structure deforms slightl, we can use original geometr for entire analsis Bodies that undergo irreversible deformations due to forces Eamples: bent paper clip steel, concrete and wood structures under large deformations If a structure ehibits large elastic or inelastic deformations, geometr changes 9/21/2005 1

2 Statics orces We start with statics Determining that the various forces acting on a bod are in equilibrium. applied force reaction forces reaction force re actions of one bod on another Pushing against each other - compressive force (bodies in compression) Pulling against each other - tensile force ( bodies in tension) orces represented b arrow length of arrow = scalar magnitude direction of arrow = line of action of force orces orce Components orce is a vector quantit In these lecture notes, we will use Boldface to represent a vector. E.g., is a vector Your book uses arrows over letters to signif a vector (hard to see). orce can be replaced b and components E.g., is a vector In these lecture notes, we will use to represent the magnitude of. Your book uses or to signif the magnitude of. = cos = sin orces in 3 Dimensions In 3 dimensions, forces can be replaced b 3 components along, and z aes: z = (,, z ) = ( 1, 2, 3 ) or = 1 i + 2 j + 3 k z z z i, j and k are unit vectors in,, and z = ( z2 ) 0.5 9/21/2005 2

3 Tpes of orces Colinear Concurrent Coplanar Colinear orces orces acting along the same line of action. The magnitude of a single equivalent force is the same as the sum of the colinear forces = = 4 Concurrent orces Co-planar orces Pass through the same point in space Lie in the same plane z 1 2D Case 3D case 4 1 Transmissibilit Resolution of orces Etension of the concept of colinear forces. If a force is eerted on a rope or a cable, then each end must have an equal force if the sstem does not move. W = W If we have a set of concurrent forces, we can resolve these forces into a single force /21/2005 3

4 Resolution of orces... con t. Determine angles wrt +X ais To compute the resultant of several concurrent forces: first determine the angle of each force with respect to + ais find and components for each force; and sum colinear forces Note: You must be sstematic about the angles. I will show ou one sstem Note: b measuring all angles from + ais, sines and cosines will reflect whether and are positive or negative. Determine orce Components Determine X and Y components for each force 2 2 sin cos 2 1 sin 1 3 cos 3 3 sin 3 1 cos 1 = sin (+) = cos (+) = sin (+) = cos ( ) 3 Sum colinear forces in and Determine Magnitude and Direction of Resultant orce 2 cos 2 3 cos 3 1 cos 1 Then compute the resultant force: = 1 cos cos cos 3 Y = 1 sin sin sin 3 2 sin 2 1 sin 1 magnitude and direction = ( 2 + Y2 ) 0.5 = tan -1 ( Y / ) Note: all forces are summed because the sines and cosines will indicate whether the component is in the positive or negative direction. 3 sin 3 9/21/2005 4

5 Spreadsheet Solution Moments and Couples Problem 9.11 orce Mag Theta Sin Cos cos(theta) sin(theta) lb deg lb lb B C D E moment about a point is defined as the product of a force magnitude and the perpendicular distance from the force line of action to that point. Moment Magnitude, M = *d Resultant Magnitude of orce = 69.1 lbs ngle of the Resultant orce = -7.9 degrees CME 1 Ton orce, perpendicular distance 9/21/2005 5

6 Moment Eamples Moments, con t. Wall d d Wall d Moments eist even when rotation is being resisted B convention, counter clockwise moments are positive The total moment about a point is the sum of the individual moments Door Hinge Moments, con t. The following ield identical moments about : 30 kn 10 kn Eample O sin φ Moment = + O sinφ 7.5 kn φ Θ 0 m 1 m 2 m 3 m 4 m O Couples Couples are pairs of forces acting in opposite directions and separated b a distance d. Moment = * d couple arm, d What is the total moment about corner? 200 N C D 45º 6 m b 6m square B 9/21/2005 6

7 What is the total moment about corner? ( ) N 200 N C D 3m 100 (sin 45º) N 3m B 6 m b 6m square 6m Summar orces are vectors. To add forces: ou must decompose into components add magnitudes of colinear components called resolution Moments are caused b forces acting at a perpendicular distance from a point Moment = (200 - ) * 6m + (100 sin 45º N) * 3m - () * 6m = 371 Nm 9/21/2005 7