Fundamentals of Statics and Dynamics - ENGR 3340 Professor: Dr. Omar E. Meza Castillo omeza@bayamon.inter.edu http://facultad.bayamon.inter.edu/omeza Department of Mechanical Engineering

Tentative Lectures Schedule Topic Lecture Reduction of a Single Distributed Loading 10

Topic 10: Reduction of a Single Distributed Loading One thing you learn in science is that there is no perfect answer, no perfect measure. A. O. Beckman Topic 10: Reduction of a Single Distributed Loading Geometric Center

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 specific axis To define the moment of a couple To present methods for determining the resultants of nonconcurrent force systems To indicate how to reduce a simple distributed loading to a resultant force having a specified location

Reduction of a Simple Distributed Loading. Applications Is it possible to reduce this force system to a single force that will have the same external effect? If yes, how? A distributed load on the beam exists due to the weight of the lumber.

Reduction of a Simple Distributed Loading. Applications The sandbags on the beam create a distributed load. How can we determine a single equivalent resultant force and its location?

Reduction of a Simple Distributed Loading. In many situations a surface area of a body is subjected to a distributed load. Such forces are caused by winds, fluids, or the weight of items on the body’s surface. We will analyze the most common case of a distributed pressure loading. This is a uniform load along one axis of a flat rectangular body. In such cases, w is a function of x and has units of force per length.

Reduction of a Simple Distributed Loading. Magnitude of the Resultant Force Consider an element of length dx. The force magnitude dF acting on it is given as dF = w(x) dx The net force on the beam is given by +  Here A is the area under the loading curve w(x).

Reduction of a Simple Distributed Loading. Magnitude of the Resultant Force The total moment about point O is given as + or + Comparing the last two equations, we get You will learn later that FR acts through a point “C,” which is called the geometric center or centroid of the area under the loading curve w(x).

Application Problem Answers: 1. D 2. B Given: The loading on the beam as shown. Find: The equivalent force and its location from point A Plan: 1) Consider the trapezoidal loading as two separate loads (one rectangular and one triangular). 2) Find FR and for each of these two distributed loads. 3) Determine the overall FR and for the three point loadings. Answers: 1. D 2. B

Application Problem Answers: 1. D 2. B 1) For the rectangular loading of height 0.5 kN/m and width 3 m: (+ ) 2) For the triangular loading of height 2 kN/m and width 3 m: (+ ) 3 kN 1.5 kN 1.5 kN 3) No change for single force. 1 m Answers: 1. D 2. B 4) For the combined loading of the three forces: 1.5 m 4 m (+ ) xR = 1.88 m FR = 6 kN

Answers: 1. D 2. B

Answers: 1. D 2. B

Answers: 1. D 2. B

Answers: 1. D 2. B

Application Problems

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