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# CAE210 Concepts CAE210

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This 6 page Study Guide was uploaded by Yael Benarroch on Thursday January 28, 2016. The Study Guide belongs to CAE210 at University of Miami taught by Diana Arboleda in Winter 2016. Since its upload, it has received 25 views. For similar materials see Mechanics of Solids in Engineering and Tech at University of Miami.

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Date Created: 01/28/16

CAE 210 CONCEPTS First Law (when ΣF=0): An object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction if the sum of the forces acting on it are zero Second Law (when ΣF≠0, ΣF = ma): Object accelerates if the sum of the forces acting on it are not zero Third Law: For every action there is an equal and opposite reaction International System of Units (SI): The basic units are length, time, and mass which are arbitrarily defined as the meter (m), second (s), and kilogram (kg). Force derives as Newtons. U.S. Customary Units: The basic units are length, time, and force which are arbitrarily defined as the foot (ft), second (s), and pound (lb). Mass is the derived unit, as slug (lb/ft.sec) Scalar: quantity described by its magnitude only. Represented by a number-unit combination Independent of the reference coordinate system. Vector: quantity described by its magnitude AND direction Represented by an arrow and a number-unit combination Dependent of the reference coordinate system Unit Vector: magnitude always one. Used to represent direction. Components are projections of the vector into the axis of the coordinate system Concurrent forces: A set of forces which all pass through the same point. A set of concurrent forces applied to a particle may be replaced by a single resultant force which is the vector sum of the applied forces. An absolute position vector is the distance and direction of a point from the origin of the reference coordinate system. A relative position vector defines the position of one object in relation to another object. The angle between a vector and each of the axis is called a direction cosine. Direction cosines are the components of the unit vector Can be determined from position vectors Direction cosines are not independent because: cos(theta)x+cos(theta)y+cos(theta)z=1 Dot product: Applications: Compute angle between two vectors Determine component of a vector parallel and perpendicular to a given line Used to evaluate moments of force about a line A force is a push or pull influence on an object that has a tendency to produce movement (translation along a line) and/or change shape. It results from the object's interaction with another object. In physics, it is represented by a vector. To define a force: 1. Point of application (coordinates x, y, z) 2. Magnitude 3. Direction: line of action or sense (arrow tip) Springs A spring is an elastic element that deforms under the action of forces. Springs come in various shapes and sizes. Even seemingly rigid structural elements can be considered springs as they deform (to a much lesser extent) under the action of forces. When a tensile (pulling) force is applied, a spring stretches or extends. A compressive (squeezing) force decreases the length. When the spring is unloaded (force is removed), it returns to its original length. This property of returning to its original shape is known as elasticity. The plot of force vs deformation (change in length) is linear. The slope is a constant called the spring constant and it depends on the material and design of the spring. The spring constant defines the stiffness of a spring. F=kδ Free Body Diagram: A graphical representation of all the forces acting on an object. Shows the forces acting on a body and allow for the application of the equilibrium equation. Is a sketch showing only the body of our interest, isolated from its surroundings, while capturing all the forces acting on it. 1. Sketch the overall shape – object freed from supports, ropes, etc. 2. Define a reference coordinate system 3. Show ALL forces acting on that body with direction and magnitude. 4. Supports and constraints that prevent motion are also forces 5. Specify relevant dimensions MOMENT: An unbalanced force applied to an object will cause it to move Meaning the object will shift from one place to another (called translation or translational motion). An unbalanced force acting at a distance from a pivot (point) will cause it to turn. We say that this force produces a moment about that point This is called rotational motion Moment = force x perpendicular distance. Positive moments are counterclockwise Cross product: The cross product a x b of two vectors a and b, unlike the dot product, is a vector. For this reason, it is also called the vector product. It is defined only in three-dimensional (3-D) space. It is perpendicular to both a and b. Moment = force x perpendicular distance M = Fd. The position vector is the distance from the point of rotation to the point of application The moment vector is the cross product of the position vector (distance from point of rotation to point of application) and the force vector Moments about a point: A force will produce a tendency for a body to rotate about a point that is not on the line of action of the force. This tendency to rotate is sometimes called a torque, or a moment of the force, but usually, it’s just called a moment. The Moment caused by a force is a measure of its tendency to cause a body to rotate about a specific point or axis. This is different from the tendency for a body to move, or translate, in the direction of the force. To increase the rotational effect, increase either the force F or the distance d. A twisting moment is the moment applied along the axis of a structural member. Symbol T refers to the twisting moment. Torsion and torque are other commonly used terms to refer to the twisting moment. A bending moment acts normal to the axis of the structural element and tends to bend the structural element. Leverage - Increase in force gained by using a lever - Mechanical advantage gained by using a lever Lever: a strong bar used to lift or move something heavy (crowbar). The lever uses the concept of moment to gain a mechanical advantage. A couple is defined as two forces of equal magnitude, opposite directions, but different lines of action. A couple moment is a free vector Meaning it can act at any point. In a two-dimensional space, an object can translate along the x- and y- directions, and can rotate about the z-axis. The way an abject can move is called a degrees of freedom: a direction in which independent motion can occur When the number of unknown forces is more than the number of independent equations, you have a: Statically indeterminate system. TRUSSES: structure made of slender structural elements that take loads only in the form of compression or tension. Why Trusses? efficient, rapid construction, light weight Parts of a truss: - Members (2-force) - Joints (pin) - Supports (pins and rollers) – No moments In a simple truss,m = 2n - 3 where m is the total number of members and n is the number of joints. Assumptions in a truss: All loads are applied at the joints: The loads are applied only at the joints. Neglect the self-weight of the members. Zero-Force Members : Prevent buckling in the long structural members by increasing the rigidity in the transverse direction. Become redundant members and come into play when normally load-carrying structural elements fail. Support loads during construction. TWO RULES: 1. At a joint where three members meet, two of the members are collinear, and if there is no external load, the non-collinear member is a zero-force member. 2. At a joint where two members meet and there is no external load, the two members are zero-force members. DISTRIBUTED LOADS: Described by the intensity of the force per unit length of the beam The effect of a distributed load on a structure is given by: The resultant and the point of application CENTROID: point that locates the geometric center of an area Defined as the average position along its coordinate axes Center of gravity: point where the resultant moment caused by the distribution of weight is zero. SHEAR AND MOMENT DIAGRAMS: Shear force and bending moment diagrams graphically represent the magnitude of the shear force and bending moment at successive points along the length of the beam. These values are useful in calculating the stresses and estimating the possible failures. Observations: • Shear is constant between concentrated loads • (Bending) moment varies linearly between concentrated loads • Discontinuities occur in S and slope of M at concentrated loads • change in shear equals amount of concentrated load at point of application • Values of S and M (and F) go to values of reactions at boundaries • Shear varies linearly over constant distributed load • Moment varies as a square function over linear shear region These observations and relationships between Loading, Shear and Moment indicate “there may be more there”.

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