ARCH 461 - Structures Midterm 1

Resolution of a Force into a Force and Couple Acting at Another Point

(ex: The point of application of a force F acting on a rigid body can be moved to another point O by adding two additional forces F and -F -or a moment force-at the point O, which basically means that the force F may be moved to a new point provided that a couple is added with magnitude Fd - which is a moment force)

Resultant of Two Parallel Forces

(two parallel forces can be resolved into a single resultant that produces the same effect on a body as the two original forces - the magnitude of the resultant force is obtained by the algebraic summation of the two forces - the point of application of the resultant force is obtained such that the moment of the resultant about any and all points is equal to the sum of the moments of the two original points about the same point)

Force Relationships

-Collinear (same line of action) -Coplanar (all forces acting in the same plane) -Concurrent (intersecting at common point) -Parallel (do not lie in the same plane and do not cross) (various combinations of these)

Guest Lecture

-Normon Foster (High Tech Modern Design, Structure + Social/Economical Effects) - Anthony Hunt => joined forces - had a clear structural component to their designs) ex: "Democratic pavilion" - providing flexibility within single volume (simple form - large open space with movable partitions = very flexible space) - Section perspective (something they did a lot in their designs to showcase their work) - shows structure light and organization - Fabrication on-site vs off-site (cost vs time vs look) - Primary framing vs secondary framing - designed for immediate need but also anticipated for future (able to remodel without effecting workers or original structure function as a factory - I-beams extending beyond frame) -corrugated metal partition - moment frame (used cross bracing but made it artistic using shadows - one goes into tension and other goes into compression)

Analysis Models

1. Define the space (2D or 3D) 2. Specify node locations (cartesian coordinates, ID Number, Number Pattern) 3. Specify Elements (Connect to appropriate nodes, Specify End Releases, Node Stability (cannot be released on all sides), and specify section properties) 4. Specify Nodal Restraints (Set appropriate node DOF to Zero) 5. Specify Applied External Forces (Concentrated Loads on Nodes, Concentrated Loads at Specific Locations on Elements, Distributed Loads on Elements)

Newtons Laws

1. Equilibrium (dual forces or three-force) 2. The time rate of change of momentum is equal to the force producing it, and the change takes place in the direction in which the force is acting (F = ma) 3. For every action there is an equal and opposite reaction acting along the same line of action (for translational force)

Hierarchy of Structure Forces

1. External Structure Forces (applied forces - including self weight - and reactions) 2. Internal Structure Forces (general term referring to the network of forces inside a structure - results of applied loads, geometry, structural configuration, types of connections, and the relative stiffness of members) 3. External Member Forces (all forces acting on a specific member: end forces, forces at connections, external applied forces acting on that member) 4. Internal Member Forces (axial, shear, moment, torsion) 5. Internal Member Stresses (Stress = Force/Area)

Characteristics of a force

1. Point of application 2. Magnitude (strength of the force) 3. Direction

The process of structural analysis

1. REAL STRUCTURE (existing or imaginary structure being designed) 2. IDEALIZED PHYSICAL MODEL (Assumptions: theory, materials, idealization, construction, linear elastic) 3. MATHEMATICAL MODEL 4. ANALYSIS (on computer usually - equation solving) 5. RESULTS (numerical forces and graphical displacements) 6. PICTURE OF BEHAVIOR (Understanding of the behavior of the idealized model 7. REAL BEHAVIOR (compare to assumptions - check results - interpretation)

Types of force systems

1. Resolution of Forces into Rectangular Components (the resolution of a vector into two perpendicular components: X and Y) 2. Vector addition by the Component Method (vectors may be added graphically y using the parallelogram law or the tip-to-tail method) 3. Moment of a Force (M = F x d) (when a force produces a rotation of a body about some reference axis or point - defined as the product of the magnitude of the force F and the perpendicular distance d from the point to the line of action) 4. Varignon's Theorem (the moment of a force about a point (axis) is equal to the algebraic sun of the moments of its components about the same point (axis)). M=(Fx x dy) - (Fy x dx) subtraction is because the force Fy is going down 5. Couple and Moment of a Couple (two forces have the same magnitude and parallel lines of action, but opposite directions) M = -Fd where F is the forces of the couple (the same but opposite) and d is the distance between the couples (so the distance of the couple from the point does not matter) 6. Resolution of a Force into a Force and Couple Acting at Another Point (ex: The point of application of a force F acting on a rigid body can be moved to another point O by adding two additional forces F and -F -or a moment force-at the point O, which basically means that the force F may be moved to a new point provided that a couple is added with magnitude Fd - which is a moment force) 7. Resultant of Two Parallel Forces (two parallel forces can be resolved into a single resultant that produces the same effect on a body as the two original forces - the magnitude of the resultant force is obtained by the algebraic summation of the two forces - the point of application of the resultant force is obtained such that the moment of the resultant about any and all points is equal to the sum of the moments of the two original points about the same point)

Particles

A point in space: Infinitely Stiff and Strong with NO DIMENSION

Vectors

A quantity having direction as well as magnitude, especially as determining the position of one point in space relative to another.

Free Body Diagrams (FBD)

A schematic representation (model) of a body removed from its supports or from a larger body. A complete (solved) FBD must be in equilibrium and all externally applied forces and reactions must be shown. The usefulness of FBD's comes from the fact that whenever we construct one we are, by definition, making cuts. MAKING CUTS AND SOLVING FOR UNKNOWN FORCES IS THE PRINCIPAL TOOL OF STATICS

Multiframe Software

Analysis Model - nodes (rigid bodies - no dimension) - member - hinge (released member) vs welded connections - pin vs fixed connections

Elasticity, Hooke's Law, and Young's Modulus

Elasticity: the ability of an object or material to resume its normal shape after being stretched or compressed; stretchiness - it is a relationship between stress and strain Hooke's law is a principle of physics that states that the force F needed to extend or compress a spring by some distance X is proportional to that distance. That is: F = kX, where k is a constant factor characteristic of the spring: its stiffness, and X is small compared to the total possible deformation of the spring. Young's Modulus: the slope of the straight line portion of the stress-strain diagram

Sign convention of moments

Following the right hand rule, for an axis coming towards you out of the page: counterclockwise is POSITIVE clockwise is NEGATIVE

Engineering materials and their properties

Frame/Bending Structures Stacking/Heavy Structures Tension (Tent-Canvas) - Limited until 19th c. (steel cables) Umbrella (Membrane Structure) Basket Structure (Frame) Magnitude vs. Taking Up Space Same Load, Bigger Area = Lower Stress More Surface Area of support = Better Support

Rigid Bodies

Infinitely Stiff and Strong (no deformation) Has dimension (theoretical body of matter)

Equilibrium of Compound Beams

One beam on top of another needs to be separated (using FBD and idea of equilibrium in each separate piece) and use point of intersection as equal and opposite forces to move from one piece to the other

Cuts and Internal Member Forces

P = axial force V = Shear force (rotation clockwise is positive - sheer force amount depends on applied force amount and is continuous until reaching that force where it then jumps up or down the amount of that force (where as before it was the amount of the resisting force)) Sheer follows the direction of the applied force usually but the positive or negative depends on movement of sheer force and resistant force (ex if they move in clockwise rotation = positive sheer) in the case of a distributed force the sheer would be sloping down (the slope = w) M = Moment force (positive = concave - smiley) (negative = convex - frowny) usually has a max moment at point of applied force

Forces

Represents the action of one body on another (understood in terms of Newton's three laws) 1. Equilibrium (dual forces or three-force) 2. Acceleration is produced when a force acts on a mass (the greater the mass, the greater amount of force needed) 3. For every action there is an equal and opposite reaction acting along the same line of action (for translational force) Characterized by three things: 1. Point of application 2. Magnitude (strength of the force) 3. Direction We can represent forces as VECTORS (force displacement) acting at a particular point on a structure

Supports and Connections

Roller (reaction on y-axis if horizontal roller and x-axis if vertical roller) Pin (reaction in x-axis and y-axis) Fixed (reaction in -x-axis, y-axis, and moment) Simple (on frictionless surface - we havent gone over this yet)

Principle of Transmissibility

States that the condition of equilibrium of a rigid body will remain unchanged if a force acting at a given point is replaced by a force of the same magnitude and direction, but acting at a different point ALONG ITS LINE OF ACTION. We can think of this as moving a given force anywhere along its line of action without changing its effect on a rigid body.

Engineering requirements of structure

Strength Stiffness Stability

Stress and Strain

Stress = a term used to describe the intensity of a force - the quantity of force that acts on a unit of area: Stress = Force/Area Strain = the deformation per unit length (the result of a change of temperature or stress): Strain = total deformation/original length

Architectural requirements of structure

Support Span Brace

Moment of a couple

The final moment M is called the moment of the couple (also independent of the location and the reference point A)

Stability

The fundamental concept of stability and equilibrium is concerned with the balancing of forces to ensure that a building and its components will not move. If you have a failure of stability, you have a problem of balance (equilibrium) and that is when a structure fails by breaking or bending too much.

Kinematic Indeterminacy

The number of unknown degrees of freedom (displacements) ex: if 11 Unknown displacements it is Kinematically Indeterminate to the 11th degree

Parallelogram Law for the addition of forces

Two concurrent forces can be replaced by a single force (the RESULTANT) obtained by drawing the diagonal of the force parallelogram.

Couple

Two forces have the same magnitude and parallel lines of action, but opposite directions M = -Fd where F is the forces of the couple (the same but opposite) and d is the distance between the couples (so the distance of the couple from the point does not matter)

Degrees of Freedom

Unless restrained, entities in 3D Cartesian space are free to move and rotate (displace) in any direction - 6 potential degrees of freedom (3 translational and three rotational) In 2D space there are 3 DOF (2 translational and 1 rotational about the z axis) - depending on the type of connection (if any) those DOF will be limited (ex: pin connection in 2D can only have DOF rotation about the z axis)

Sign conventions for forces

Up and to the right is POSITIVE Down and to the left is NEGATIVE

Equilibrium

When the net effect of all forces = 0 (equations: sum of Fy = 0, sum of Fx = 0, sum of M=0) Balanced (no movement) Equal and Opposite

Resultants of distributed forces

distributed forces = distributed loads R = wl where l is the length of the distributed load and w is the weight per unit distance of the distributed load The resultant of a distributed load acts in the middle of that load

Varignon's Theorem

the moment of a force about a point (axis) is equal to the algebraic sun of the moments of its components about the same point (axis) M=(Fx x dy) - (Fy x dx) subtraction is because the force Fy is going down