🔗Statics and Strength of Materials Unit 2 – Force Systems and Resultants
Force systems and resultants form the foundation of statics. This unit covers the basics of forces, their types, and how they interact. You'll learn about vector analysis, free body diagrams, and equilibrium conditions.
Understanding resultant forces and moments is crucial for simplifying complex force systems. This knowledge applies to real-world structures like trusses, frames, and beams. Mastering these concepts will help you analyze and design various mechanical and structural systems.
Force defined as a push or pull that acts upon an object resulting from the object's interaction with another object
Magnitude, direction, and point of application must be specified to fully describe a force
Force is a vector quantity, having both magnitude and direction
Concurrent forces lines of action intersect at a common point
Coplanar forces lie in a common plane
Collinear forces lines of action lie on the same line
Can be replaced by a single equivalent force
Transmissibility of a force states that the point of application of a force can be moved along its line of action without changing its effect on the body
Types of Force Systems
Parallel force system consists of forces with lines of action that are parallel to each other
Can be concurrent (lines of action intersect) or non-concurrent (lines of action do not intersect)
General force system contains forces that are neither concurrent nor parallel
Coplanar force system all forces act in a single plane
Three-dimensional (3D) force system forces act in three dimensions, not confined to a single plane
Couple two equal, opposite, and non-collinear forces that produce a pure moment (rotational effect)
Wrench a combination of a force and a couple acting on a body
Distributed load a load that is spread out over an area (pressure) or along a line (line load)
Vector Analysis of Forces
Forces can be represented as vectors, with magnitude and direction
Vector addition used to find the resultant force of multiple forces acting on a body
Parallelogram law used for graphical vector addition
Triangle rule used for graphical vector addition of three or more forces
Rectangular components breaking a force vector into its x and y components using trigonometry
Allows for easier mathematical analysis of forces
Moment of a force the turning effect of a force about a point, calculated as the cross product of the position vector and the force vector M=r×F
Varignon's theorem states that the moment of a force about a point is equal to the sum of the moments of its rectangular components about the same point
Free Body Diagrams
Free body diagram (FBD) a simplified representation of an object showing all external forces acting on it
Helps to visualize and analyze force systems
Steps to draw an FBD:
Isolate the object of interest
Replace supports or connections with appropriate reaction forces and moments
Draw all external forces acting on the object
Label each force with its magnitude, direction, and point of application
Types of supports and their reactions:
Roller support provides a single force perpendicular to the surface
Pin support provides both horizontal and vertical force components
Fixed support provides both force and moment reactions
Distributed loads represented as equivalent concentrated forces in FBDs for simplicity
Equilibrium Conditions
Equilibrium a state in which an object is at rest or moving with constant velocity, with no net force or moment acting on it
First condition of equilibrium: The sum of all forces acting on an object must be zero ∑F=0
For 2D equilibrium, both x and y components of forces must sum to zero ∑Fx=0 and ∑Fy=0
For 3D equilibrium, x, y, and z components must sum to zero ∑Fx=0, ∑Fy=0, and ∑Fz=0
Second condition of equilibrium: The sum of all moments about any point must be zero ∑M=0
For 2D equilibrium, moments about a single point must sum to zero ∑MO=0
For 3D equilibrium, moments about three non-collinear points must sum to zero ∑MA=0, ∑MB=0, and ∑MC=0
Static determinacy a system is statically determinate if all unknown forces and moments can be found using equilibrium equations alone
Requires that the number of equilibrium equations is equal to the number of unknowns
Resultant Forces and Moments
Resultant force a single force that has the same effect on a body as a system of forces
Found using vector addition of all forces in the system
Resultant moment a single moment that has the same effect on a body as a system of moments
Found by summing all moments about a chosen point
Equivalent force systems two force systems that have the same resultant force and moment
Can be used to simplify complex force systems for analysis
Reduction of a force system the process of finding an equivalent force-couple system at a chosen point
Involves calculating the resultant force and moment at the chosen point
Equipollent force systems have the same resultant force and the same resultant moment about any point
Practical Applications
Trusses composed of two-force members connected at joints, used in bridges, roofs, and cranes
Method of joints used to analyze forces in truss members by applying equilibrium at each joint
Method of sections used to analyze forces in truss members by applying equilibrium to a section of the truss
Frames and machines composed of multi-force members, used in vehicles, furniture, and gym equipment
Static analysis involves applying equilibrium conditions to each member and joint to determine unknown forces and moments
Beams structural elements that primarily resist bending, used in buildings, bridges, and machinery
Shear and moment diagrams used to visualize internal shear forces and bending moments along the length of the beam
Friction forces resist relative motion between surfaces in contact
Coulomb friction model used to calculate friction force based on normal force and coefficient of friction Ff≤μN