CH04_101_Moments_Cross_Products_Handouts

Page 1: Introduction to Moments of Forces

• Moment of a Force

  • Definition: A measure of the tendency of a force to cause a body to rotate about an axis.

  • Applications in Civil Engineering:

    • Design of retaining walls.

    • Analysis of dams.

    • Consideration of wind and earthquake forces as applied loads.

  • Example: A wrench turning a pipe:

    • Increasing force (Fx) and/or distance increases torque (moment of force).

    • Torque direction and effectiveness depend on force line of action relative to the particle.

    • Forces in y-direction (Fy) and z-direction (Fz) do not cause rotation about the z-axis.

Page 2: Understanding Magnitude and Direction of Moments

• Magnitude of a Moment:

  • Formula: ( M = Fd )

    • d = moment arm = perpendicular distance from moment axis to force.

    • Units: N-m (Newton-meters) or lb-ft (foot-pounds).

• Direction of a Moment:

  • Right-Hand Rule: Fingers in direction of force, thumb shows moment direction.

  • Positive moments: Counterclockwise.

  • Negative moments: Clockwise.

• Resultant Moment in 2D:

  • Directed along the z-axis:

    • Counterclockwise moments are positive.

    • Clockwise moments are negative.

    • Resultant moment = algebraic sum of all individual moments.

Page 3: Scalar Formulation of Moment of a Force

• Moment of a Force - Scalar Formulation:

  • MO indicates tendency for rotation around point O.

    • In 2D, MO direction is either clockwise or counterclockwise based on rotation tendency.

  • Formula: ( Mo = Fd )

  • Example: Calculating moments with given coordinates and force.

Page 4: Additional Moment Calculations

• Continued Examples:

  • Moments can be calculated for various points in space, considering both position and force vector components.

  • Vector Components:

    • MO can be determined using force components:

      • ( MO = (Fy * a) - (Fx * b) )

      • Use direction conventions (counterclockwise positive).

Page 5: Cross Product and Moments in 3D

• Cross Product in 3D:

  • Method to find moments of forces in three dimensions.

  • Vector operation of cross product helps determine moments and is defined as:

    • ( C = A × B )

    • Magnitude and direction of ( C ) are pivotal for analyses.

  • Right-Hand Rule used for direction.

  • Not commutative: ( A × B ≠ B × A )

Page 6: Calculating Cross Products

• Unit Vector Relationships:

  • Directions represented by unit vectors (i, j, k).

  • Basic results of cross products between unit vectors provided (e.g., ( \hat{i} × \hat{j} = \hat{k} )).

• Determinants for Cross Products:

  • Expand cross product using matrices for clarity and ease of computation.

Page 7: Moment Calculation Using Vectors

• Moment with Respect to Origin:

  • Moment derived from vector cross product using position vector from the origin:

    • ( MO = r × F )

  • Examples demonstrate this relationship with given force vectors and coordinates.

Page 8: Resultant Moments from Multiple Forces

• Resultant Moments:

  • The resultant of multiple moments is simply the algebraic sum of each individual moment created by forces acting.

  • Positive and negative signs indicating direction (counterclockwise vs. clockwise).

Page 9: Maximum Moments and Angle Considerations

• Force Angles Impacting Moments:

  • Maximum moments occur when angles yield optimal force applications relative to rotation axes.

  • Calculation of moments situating forces at various points regarding axes is key to analysis.

Page 10: Moment Projections on Specified Axis

• Methods for Moment Calculation:

  • Two approaches:

    1. Find component of the moment about the specific axis.

    2. Calculate moment first, then project onto the axis of interest.

  • General expressions derived for clarity.

Page 11: Understanding Couples and Moments

• Couple Moments:

  • Defined by two parallel forces of equal magnitude but opposite directions, separated by a distance.

  • Generate a moment without a net force, affecting rotation alone.

• Example of Couples: Moment’s independence from the choice of reference axis.

Page 12: Further Concepts on Couples

• 2D and 3D Couples:

  • Scalar and vector representations provided with emphasis on distance and direction using the right-hand rule.

  • Equivalent couples: Force couples yield identical moments irrespective of their application point on a body.

Page 13: Resultant Moments from Force Systems

• Equivalent Force and Moment Representation:

  • Describe systems subjected to multiple forces and moments using single resultant forces and moments for convenience in analysis.

Page 14: Moving Forces in Analysis

• Principle of Transmissibility:

  • A force can be moved along its line of action without altering the body's reaction to external effects. This confirms the nature of forces as sliding vectors along their lines.

Page 15: Resultants from Force Systems Simplified

• Reduction of Systems:

  • Analyzing forces and moments from a system helps in representing their combined effect at a single reference point.

  • Free vector properties of moment aids in simplification.

Page 16: Additional Examples and Analysis

• Example Calculations: Applying learned principles to specific examples.

  • Understanding resultant forces and moments configuration.

Page 17: Distributed Load Analysis

• Distributed Loading Principles:

  • Understanding pressure at various points on a surface and resultant conditions via single force equivalents.

• Uniform vs Non-Uniform Loading:

  • Discuss methods of determining resultant forces from both uniform and variable distributed loads.

Page 18: Further Cases of Distributed Loads

• Examples: Addressing theoretical loads and their results on systems, emphasizing centroids in various configurations for resultant loading.

Page 19: Review and Simplification Techniques

• Last Examples and Summarizations: Conclude with practical examples and methods to reduce complex loading scenarios into manageable forces and moments.