Engineering Mechanics: Statics

Name: Engineering Mechanics: Statics
Uploaded: 2017-07-15T16:53:54+00:00
Duration: PTM38S49
Channel: Willa Malone
Description: Engineering Mechanics: Statics

Engineering Mechanics: Statics
Chapter 4: Force System Resultants Engineering Mechanics: Statics

Chapter Objectives To discuss the concept of the moment of a force and show how to calculate it in two and three dimensions. To provide a method for finding the moment of a force about a specified axis. To define the moment of a couple. To present methods for determining the resultants of non-concurrent force systems. To indicate how to reduce a simple distributed loading to a resultant force having a specified location.

Chapter Outline Moment of a Force – Scalar Formation Cross Product
Moment of Force – Vector Formulation Principle of Moments Moment of a Force about a Specified Axis

Chapter Outline Moment of a Couple Equivalent System
Resultants of a Force and Couple System Reduction of a Simple Distributed Loading

4.1 Moment of a Force – Scalar Formation
Moment of a force about a point or axis – a measure of the tendency of the force to cause a body to rotate about the point or axis Case 1 Consider horizontal force Fx, which acts perpendicular to the handle of the wrench and is located dy from the point O

Fx tends to turn the pipe about the z axis The larger the force or the distance dy, the greater the turning effect Torque – tendency of rotation caused by Fx or simple moment (Mo) z

Moment axis (z) is perpendicular to shaded plane (x-y) Fx and dy lies on the shaded plane (x-y) Moment axis (z) intersects the plane at point O

Case 2 Apply force Fz to the wrench Pipe does not rotate about z axis Tendency to rotate about x axis The pipe may not actually rotate Fz creates tendency for rotation so moment (Mo) x is produced

Case 2 Moment axis (x) is perpendicular to shaded plane (y-z) Fz and dy lies on the shaded plane (y-z)

Case 3 Apply force Fy to the wrench No moment is produced about point O Lack of tendency to rotate as line of action passes through O

In General Consider the force F and the point O which lies in the shaded plane The moment MO about point O, or about an axis passing through O and perpendicular to the plane, is a vector quantity Moment MO has its specified magnitude and direction

Magnitude For magnitude of MO, MO = Fd where d = moment arm or perpendicular distance from the axis at point O to its line of action of the force Units for moment is N.m

Direction Direction of MO is specified by using “right hand rule” - fingers of the right hand are curled to follow the sense of rotation when force rotates about point O

Direction - Thumb points along the moment axis to give the direction and sense of the moment vector - Moment vector is upwards and perpendicular to the shaded plane

Direction MO is shown by a vector arrow with a curl to distinguish it from force vector Example (Fig b) MO is represented by the counterclockwise curl, which indicates the action of F

Direction Arrowhead shows the sense of rotation caused by F Using the right hand rule, the direction and sense of the moment vector points out of the page In 2D problems, moment of the force is found about a point O

Direction Moment acts about an axis perpendicular to the plane containing F and d Moment axis intersects the plane at point O

Resultant Moment of a System of Coplanar Forces Resultant moment, MRo = addition of the moments of all the forces algebraically since all moment forces are collinear MRo = ∑Fd taking clockwise to be positive

Resultant Moment of a System of Coplanar Forces A clockwise curl is written along the equation to indicate that a positive moment if directed along the + z axis and negative along the – z axis

Moment of a force does not always cause rotation Force F tends to rotate the beam clockwise about A with moment MA = FdA Force F tends to rotate the beam counterclockwise about B with moment MB = FdB Hence support at A prevents the rotation

Example 4.1 For each case, determine the moment of the force about point O

Solution Line of action is extended as a dashed line to establish moment arm d Tendency to rotate is indicated and the orbit is shown as a colored curl

Solution

Example 4.2 Determine the moments of the 800N force acting on the frame about points A, B, C and D.

Solution Scalar Analysis Line of action of F passes through C

4.2 Cross Product Cross product of two vectors A and B yields C, which is written as C = A X B Read as “C equals A cross B”

4.2 Cross Product Magnitude
Magnitude of C is defined as the product of the magnitudes of A and B and the sine of the angle θ between their tails For angle θ, 0° ≤ θ ≤ 180° Therefore, C = AB sinθ

4.2 Cross Product Direction
Vector C has a direction that is perpendicular to the plane containing A and B such that C is specified by the right hand rule - Curling the fingers of the right hand form vector A (cross) to vector B - Thumb points in the direction of vector C

4.2 Cross Product Expressing vector C when magnitude and direction are known C = A X B = (AB sinθ)uC where scalar AB sinθ defines the magnitude of vector C unit vector uC defines the direction of vector C

4.2 Cross Product Laws of Operations 1. Commutative law is not valid
A X B ≠ B X A Rather, A X B = - B X A Shown by the right hand rule Cross product A X B yields a vector opposite in direction to C B X A = -C

4.2 Cross Product Laws of Operations 2. Multiplication by a Scalar
a( A X B ) = (aA) X B = A X (aB) = ( A X B )a 3. Distributive Law A X ( B + D ) = ( A X B ) + ( A X D ) Proper order of the cross product must be maintained since they are not commutative

4.2 Cross Product Cartesian Vector Formulation
Use C = AB sinθ on pair of Cartesian unit vectors Example For i X j, (i)(j)(sin90°) = (1)(1)(1) = 1

4.2 Cross Product Laws of Operations In a similar manner,
i X j = k i X k = -j i X i = 0 j X k = i j X i = -k j X j = 0 k X i = j k X j = -i k X k = 0 Use the circle for the results. Crossing CCW yield positive and CW yields negative results

4.2 Cross Product Laws of Operations
Consider cross product of vector A and B A X B = (Axi + Ayj + Azk) X (Bxi + Byj + Bzk) = AxBx (i X i) + AxBy (i X j) + AxBz (i X k) + AyBx (j X i) + AyBy (j X j) + AyBz (j X k) AzBx (k X i) +AzBy (k X j) +AzBz (k X k) = (AyBz – AzBy)i – (AxBz - AzBx)j + (AxBy – AyBx)k

4.2 Cross Product Laws of Operations In determinant form,

4.3 Moment of Force - Vector Formulation
Moment of force F about point O can be expressed using cross product MO = r X F where r represents position vector from O to any point lying on the line of action of F

Magnitude For magnitude of cross product, MO = rF sinθ where θ is the angle measured between tails of r and F Treat r as a sliding vector. Since d = r sinθ, MO = rF sinθ = F (rsinθ) = Fd

Direction Direction and sense of MO are determined by right-hand rule - Extend r to the dashed position - Curl fingers from r towards F - Direction of MO is the same as the direction of the thumb

Direction *Note: - “curl” of the fingers indicates the sense of rotation - Maintain proper order of r and F since cross product is not commutative

Principle of Transmissibility For force F applied at any point A, moment created about O is MO = rA x F F has the properties of a sliding vector and therefore act at any point along its line of action and still create the same moment about O

Cartesian Vector Formulation For force expressed in Cartesian form, where rx, ry, rz represent the x, y, z components of the position vector and Fx, Fy, Fz represent that of the force vector

Cartesian Vector Formulation With the determinant expended, MO = (ryFz – rzFy)i – (rxFz - rzFx)j + (rxFy – yFx)k MO is always perpendicular to the plane containing r and F Computation of moment by cross product is better than scalar for 3D problems

Cartesian Vector Formulation Resultant moment of forces about point O can be determined by vector addition MRo = ∑(r x F)

Moment of force F about point A, pulling on cable BC at any point along its line of action, will remain constant Given the perpendicular distance from A to cable is rd MA = rdF In 3D problems, MA = rBC x F

Example 4.4 The pole is subjected to a 60N force that is directed from C to B. Determine the magnitude of the moment created by this force about the support at A.

Solution Either one of the two position vectors can be used for the solution, since MA = rB x F or MA = rC x F Position vectors are represented as rB = {1i + 3j + 2k} m and rC = {3i + 4j} m Force F has magnitude 60N and is directed from C to B

Solution Substitute into determinant formulation

Solution Or Substitute into determinant formulation For magnitude,

4.4 Principles of Moments Also known as Varignon’s Theorem
“Moment of a force about a point is equal to the sum of the moments of the forces’ components about the point” For F = F1 + F2, MO = r X F1 + r X F2 = r X (F1 + F2) = r X F

4.4 Principles of Moments The guy cable exerts a force F on the pole and creates a moment about the base at A MA = Fd If the force is replaced by Fx and Fy at point B where the cable acts on the pole, the sum of moment about point A yields the same resultant moment

4.4 Principles of Moments Fy create zero moment about A MA = Fxh
Apply principle of transmissibility and slide the force where line of action intersects the ground at C, Fx create zero moment about A MA = Fyb

4.4 Principles of Moments Example 4.6
The force F acts at the end of the angle bracket. Determine the moment of the force about point O.

4.4 Principles of Moments Solution Method 1
MO = 400sin30°N(0.2m)-400cos30°N(0.4m) = -98.6N.m = 98.6N.m (CCW) As a Cartesian vector, MO = {-98.6k}N.m

4.4 Principles of Moments Solution Method 2:
Express as Cartesian vector r = {0.4i – 0.2j}N F = {400sin30°i – 400cos30°j}N = {200.0i – 346.4j}N For moment,

4.6 Moment of a Couple Couple - two parallel forces
- same magnitude but opposite direction - separated by perpendicular distance d Resultant force = 0 Tendency to rotate in specified direction Couple moment = sum of moments of both couple forces about any arbitrary point

4.6 Moment of a Couple Example
Position vectors rA and rA are directed from O to A and B, lying on the line of action of F and –F Couple moment about O M = rA X (-F) + rA X (F) Couple moment about A M = r X F since moment of –F about A = 0

4.6 Moment of a Couple A couple moment is a free vector
- It can act at any point since M depends only on the position vector r directed between forces and not position vectors rA and rB, directed from O to the forces - Unlike moment of force, it do not require a definite point or axis

4.6 Moment of a Couple Scalar Formulation Magnitude of couple moment
M = Fd Direction and sense are determined by right hand rule In all cases, M acts perpendicular to plane containing the forces

4.6 Moment of a Couple Vector Formulation For couple moment, M = r X F
If moments are taken about point A, moment of –F is zero about this point r is crossed with the force to which it is directed

4.6 Moment of a Couple Equivalent Couples
Two couples are equivalent if they produce the same moment Since moment produced by the couple is always perpendicular to the plane containing the forces, forces of equal couples either lie on the same plane or plane parallel to one another

4.6 Moment of a Couple Resultant Couple Moment
Couple moments are free vectors and may be applied to any point P and added vectorially For resultant moment of two couples at point P, MR = M1 + M2 For more than 2 moments, MR = ∑(r X F)

4.6 Moment of a Couple Frictional forces (floor) on the blades of the machine creates a moment Mc that tends to turn it An equal and opposite moment must be applied by the operator to prevent turning Couple moment Mc = Fd is applied on the handle

4.6 Moment of a Couple Example 4.10
A couple acts on the gear teeth. Replace it by an equivalent couple having a pair of forces that cat through points A and B.

4.6 Moment of a Couple Solution Magnitude of couple
M = Fd = (40)(0.6) = 24N.m Direction out of the page since forces tend to rotate CW M is a free vector and can be placed anywhere

4.6 Moment of a Couple Solution To preserve CCW motion,
vertical forces acting through points A and B must be directed as shown For magnitude of each force, M = Fd 24N.m = F(0.2m) F = 120N

4.7 Equivalent System A force has the effect of both translating and rotating a body The extent of the effect depends on how and where the force is applied We can simplify a system of forces and moments into a single resultant and moment acting at a specified point O A system of forces and moments is then equivalent to the single resultant force and moment acting at a specified point O

4.7 Equivalent System Point O is on the Line of Action
Consider body subjected to force F applied to point A Apply force to point O without altering external effects on body - Apply equal but opposite forces F and –F at O

- Two forces indicated by the slash across them can be cancelled, leaving force at point O - An equivalent system has be maintained between each of the diagrams, shown by the equal signs

- Force has been simply transmitted along its line of action from point A to point O - External effects remain unchanged after force is moved - Internal effects depend on location of F

4.7 Equivalent System Point O is Not on the Line of Action
F is to be moved to point ) without altering the external effects on the body Apply equal and opposite forces at point O The two forces indicated by a slash across them, form a couple that has a moment perpendicular to F

4.7 Equivalent System Point O is Not on the Line of Action
The moment is defined by cross product M = r X F Couple moment is free vector and can be applied to any point P on the body

4.8 Resultants of a Force and Couple System
Consider a rigid body Since O does not lies on the line of action, an equivalent effect is produced if the forces are moved to point O and the corresponding moments are M1 = r1 X F1 and M2 = r2 X F2 For resultant forces and moments, FR = F1 + F2 and MR = M1 + M2

Equivalency is maintained thus each force and couple system cause the same external effects Both magnitude and direction of FR do not depend on the location of point O MRo depends on location of point O since M1 and M2 are determined using position vectors r1 and r2 MRo is a free vector and can acts on any point on the body

Simplifying any force and couple system, FR = ∑F MR = ∑MC + ∑MO If the force system lies on the x-y plane and any couple moments are perpendicular to this plane, FRx = ∑Fx FRy = ∑Fy MRo = ∑MC + ∑MO

Procedure for Analysis When applying the following equations, FR = ∑F MR = ∑MC + ∑MO FRx = ∑Fx FRy = ∑Fy MRo = ∑MC + ∑MO Establish the coordinate axes with the origin located at the point O and the axes having a selected orientation

Procedure for Analysis Force Summation For coplanar force system, resolve each force into x and y components If the component is directed along the positive x or y axis, it represent a positive scalar If the component is directed along the negative x or y axis, it represent a negative scalar In 3D problems, represent forces as Cartesian vector before force summation

Procedure for Analysis Moment Summation For moment of coplanar force system about point O, use Principle of Moment Determine the moments of each components rather than of the force itself In 3D problems, use vector cross product to determine moment of each force Position vectors extend from point O to any point on the line of action of each force

Example 4.14 Replace the forces acting on the brace by an equivalent resultant force and couple moment acting at point A.

Solution Force Summation For x and y components of resultant force,

Solution For magnitude of resultant force For direction of resultant force

Solution Moment Summation Summation of moments about point A, When MRA and FR act on point A, they will produce the same external effect or reactions at the support

4.9 Reduction of a Simple Distributed Loading
Large surface area of a body may be subjected to distributed loadings such as those caused by wind, fluids, or weight of material supported over body’s surface Intensity of these loadings at each point on the surface is defined as the pressure p Pressure is measured in pascals (Pa) 1 Pa = 1N/m2

Most common case of distributed pressure loading is uniform loading along one axis of a flat rectangular body Direction of the intensity of the pressure load is indicated by arrows shown on the load-intensity diagram Entire loading on the plate is a system of parallel forces, infinite in number, each acting on a separate differential area of the plate

Loading function p = p(x) Pa, is a function of x since pressure is uniform along the y axis Multiply the loading function by the width w = p(x)N/m2]a m = w(x) N/m Loading function is a measure of load distribution along line y = 0, which is in the symmetry of the loading Measured as force per unit length rather than per unit area

Load-intensity diagram for w = w(x) can be represented by a system of coplanar parallel This system of forces can be simplified into a single resultant force FR and its location can be specified

Magnitude of Resultant Force FR = ∑F Integration is used for infinite number of parallel forces dF acting along the plate For entire plate length, Magnitude of resultant force is equal to the total area A under the loading diagram w = w(x)

Location of Resultant Force MR = ∑MO Location of the line of action of FR can be determined by equating the moments of the force resultant and the force distribution about point O dF produces a moment of xdF = x w(x) dx about O For the entire plate,

Location of Resultant Force Solving, Resultant force has a line of action which passes through the centroid C (geometric center) of the area defined by the distributed loading diagram w(x)

Location of Resultant Force Consider 3D pressure loading p(x), the resultant force has a magnitude equal to the volume under the distributed-loading curve p = p(x) and a line of action which passes through the centroid (geometric center) of this volume Distribution diagram can be in any form of shapes such as rectangle, triangle etc

Beam supporting this stack of lumber is subjected to a uniform distributed loading, and so the load-intensity diagram has a rectangular shape If the load-intensity is wo, resultant is determined from the are of the rectangle FR = wob

Line of action passes through the centroid or center of the rectangle, = a + b/2 Resultant is equivalent to the distributed load Both loadings produce same “external” effects or support reactions on the beam

Example 4.20 Determine the magnitude and location of the equivalent resultant force acting on the shaft

Solution For the colored differential area element, For resultant force

Solution For location of line of action, Checking,

Example 4.21 A distributed loading of p = 800x Pa acts over the top surface of the beam. Determine the magnitude and location of the equivalent force.

Solution Loading function of p = 800x Pa indicates that the load intensity varies uniformly from p = 0 at x = 0 to p = 7200Pa at x = 9m For loading, w = (800x N/m2)(0.2m) = (160x) N/m Magnitude of resultant force = area under the triangle FR = ½(9m)(1440N/m) = 6480 N = 6.48 kN

Solution Resultant force acts through the centroid of the volume of the loading diagram p = p(x) FR intersects the x-y plane at point (6m, 0) Magnitude of resultant force = volume under the triangle FR = V = ½(7200N/m2)(0.2m) = 6.48 kN

Example 4.22 The granular material exerts the distributed loading on the beam. Determine the magnitude and location of the equivalent resultant of this load

Solution Area of loading diagram is trapezoid Magnitude of each force = associated area F1 = ½(9m)(50kN/m) = 225kN F2 = ½(9m)(100kN/m) = 450kN Line of these parallel forces act through the centroid of associated areas and insect beams at

Solution Two parallel Forces F1 and F2 can be reduced to a single resultant force FR For magnitude of resultant force, For location of resultant force,

Solution *Note: Trapezoidal area can be divided into two triangular areas, F1 = ½(9m)(100kN/m) = 450kN F2 = ½(9m)(50kN/m) = 225kN

Engineering Mechanics: Statics

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