The Funicular and Moments

Abstract

In Chap. 3 it was shown that the funicular is a powerful tool that has multiple levels of meaning. It has a physical interpretation, namely the shape that a chain would take on due to multiple loads. That shape can then be used to identify the precise location of the resultant of these loads, by intersecting the starting and ending slopes of the funicular. But in this chapter, yet another meaning of the funicular is revealed and explained. The funicular automatically generates a bending moment diagram. This was introduced in Chap. 3. It will be helpful to summarize what a bending moment actually is and how it links to a funicular. In Fig. 4.1, consider a force F and find its moment about point O. This diagram Fig. 4.1 is the form diagram, although there is very little information other than verticality of F and distance to O.

Chapter 4 Exercises

Exercise 4.1 Draw the force polygon, and capture HForce

Exercise 4.1 solution Notice that a to b captures the applied force. The location of the pole p is arbitrary.

Exercise 4.2 Draw the force polygon, and capture HForce

Exercise 4.2 solution Notice that a to b captures the applied force. The location of the pole p is arbitrary.

Exercise 4.3 Draw the force polygon, and capture HForce

Exercise 4.3 solution Notice that the labels for magnitude of forces, and for locations of points are no longer useful. The pole position is arbitrary.

Exercise 4.4 Draw the force polygon, and capture HForce

Exercise 4.4 solution Notice that the labels for magnitude of forces, and for locations of points are no longer useful. The pole position is arbitrary.

Exercise 4.5 Draw the force polygon, and capture HForce

Exercise 4.5 solution Notice that the labels for magnitude of forces, and for locations of points are no longer useful. The pole position is arbitrary. The first ray ap captures Space A, to the left of the roller, but before the first load.

Exercise 4.6 Draw the force polygon, and capture HForce

Exercise 4.6 solution Notice that the labels for magnitude of forces, and for locations of points are no longer useful. The pole position is arbitrary. The first ray ap captures Space A, to the left of the roller, but before the first load.

Exercise 4.7 For the problem shown in 4.1, draw the bending moment and display the internal moment at any value of x.

Exercise 4.7 solution

Exercise 4.8 For the problem shown in 4.2, draw the bending moment and display the internal moment at any value of x.

Exercise 4.8 solution

Exercise 4.9 For the problem shown in 4.3, draw the bending moment and display the internal moment at any value of x.

Exercise 4.9 solution

Exercise 4.10 For the problem shown in 4.4, draw the bending moment and display the internal moment at any value of x.

Exercise 4.10 solution

Exercise 4.11 For the problem shown in 4.5, draw the bending moment and display the internal moment at any value of x.

Exercise 4.11 solution

Exercise 4.12 For the problem shown in 4.6, draw the bending moment and display the internal moment at any value of x.

Exercise 4.12 solution

Exercise 4.13 For the following beam with an overhang, change the uniform load to 10 discrete point loads. Place each point load at the centroid of 10 equally sized segments.

Exercise 4.13 solution

Exercise 4.14 For the discrete loads shown in the solution of Problem 4.13, calculate the moment in the beam over the prop support and calculate the moment near the peak positive bending cross section.

Exercise 4.14 solution

The difference between theoretical and graphical solutions can be reduced if more point loads are used to simulate the uniformly distributed load. Also, the centroidal placement of loads prevented a point load from being applied to the free end.

Exercise 4.15 For the following beam with an overhang, draw the bending moment and display the internal moment near the peak positive and above the left support to capture the peak negative moment.

Exercise 4.15 solution

The difference between theoretical and graphical solutions can be reduced if more point loads are used to simulate the uniformly distributed load.

Exercise 4.16 A beam is pinned at the left end, and it has two roller supports and an internal hinge as shown. Break up the uniform load into 6 point loads applied centroidally to the left of the hinge, and 4 point loads applied centroidally to the right of the hinge.

Exercise 4.16 solution

Exercise 4.17 Given the discretization of loads from problem 4.16, calculate the bending moment over the central roller support and at a place near the peak positive moment.

Exercise 4.17 solution Part 1

Exercise 4.17 solution Part 2

Exercise 4.18 A beam is pinned at the left end, and it has two roller supports and an internal hinge as shown. Break up the uniform load into 10 point loads applied centroidally. Then calculate the moment in the beam over the interior roller and near the region of maximum positive moment.

Exercise 4.18 solution

Exercise 4.19 A beam has a roller support at the left end, an internal hinge, and a fixed support at the right end. Calculate the bending moment directly beneath the first load F1 and at the fixed right end.

Exercise 4.19 solution

Exercise 4.20 A beam has a fixed support at its left end, an internal hinge, and a roller support. Calculate the bending moment under the left two loads and above the roller support.

Exercise 4.20 solution

Exercise 4.21 A circular, three-hinged arch is subjected to a single downward load as shown. Calculate the reactions and establish the bending moment near the area of most extreme negative bending.

Exercise 4.21 solution Part 1

Exercise 4.21 solution Part 2

Exercise 4.22 A circular, three-hinged arch is subjected to two downward loads as shown. Calculate the reactions and establish the bending moment near the area of most extreme bending.

Exercise 4.22 solution Part 1

Exercise 4.22 solution Part 2

Exercise 4.23 A three hinged circular arch is subjected to two point loads as shown. Calculate the reactions.

Exercise 4.23 solution

Exercise 4.24 A three hinged circular arch is subjected to two point loads as shown. Calculate the reactions.

Exercise 4.24 solution

Exercise 4.25 For the three hinged circular arch shown in Problem 4.22 draw the bending moment diagram and quantify the moment in several regions.

Exercise 4.25 solution

Exercise 4.26 For the three hinged circular arch shown in Problem 4.23 draw the bending moment diagram and quantify the moment in several regions.

Exercise 4.26 solution

Exercise 4.27 A three hinged circular arch is subjected to two point loads as shown. Calculate the reactions.

Exercise 4.27 solution

Exercise 4.28 For the problem shown in 4.26, show the bending moment diagram.

Exercise 4.28 solution