1. Engineering Mechanics
Equivalent force systems: problems

2. Brief Revision
Force components in rectangular coordinates and vector addition.
Cross-product and moment (torque) of a force about a point.
Dot-product between two vectors and projection of a vector along a given direction λ.
Triple-product and the moment of a force about an axis (given direction λ.)
Concept of a couple.

3. Equivalent Force Systems
Couples Mj and forces Fi act on the body.
The net forces and moments about the origin O are as shown.
Two force systems are equivalent if they have the same resultant force and moments.
Equivalent about one point O guarantees equivalence everywhere O

4. Special Force Systems
Parallel force system
Coplanar force system

5. Tugboat Problem
Four tugboats are used to bring an ocean liner to its pier. Each tugboat exerts 100kN push in the direction shown. Determine the point on hull where a single, more powerful tugboat should push to produce the same effect as the original four tugboats. Also determine the total push and its direction to be exerted by the single tugboat.

6. Tug Boats Tugboats are used to tow big liners etc. to the shore.

7. Problem 2
A concrete foundation mat in the shape of a regular hexagon with 3-m sides supports four column loads as shown. Determine the magnitude and the point of application of the resultant four loads.
Find the loads at points B and F so that the effective load passes through the center

8. Using Symmetry to convert 3D problems to 2D (adapted from http://oli
Box has 3-planes of symmetry.
Loading had only one plane of symmetry
Using symmetry and static equivalence, the problem can be converted into a 2D problem

9. Problem 3
Find the simplest resultant for the forces acting on the simply supported beam

10. Simple Example (http://oli.web.cmu.edu)

11. Distributed forces Forces act per unit volume/area/length
Per unit volume (N/m3)
Density
uniformly charged bodies (involving charge density)
Per unit area (N/m2)
hydrostatic load
wind forces
Per unit length (N/m)
Can be used when one of the dimension is large compared to the other two.
between two current carrying wires.
on the railway lines

12. Converting Force/Area to Force/Length
Placing 5 typical books, each 300mm x 200 mm x 50mm, and each weighing 10 N on a long wooden shelf that is 200 mm deep.

13. Hydrostatic Forces Act normal to the surface
Force per unit area (N/m2) (pressure)
The total pressure at depth z is:
p = ρgz = γz

14. Non-Uniform Forces (http://oli.web.cmu.edu)
Hydrostatic Forces

15. Non-Continuous non-uniform Loading
Load replaced by an equivalent triangular loading

16. Problem 4-5
An automatic valve consists of a 225 x 225 mm square plate of uniform thickness weighing 200 N (total). The valve is pivoted about a horizontal axis through A located at a distance h = 100 mm above the lower edge. Determine the depth of water d for which the valve will open.
Consider the same setup as in Problem 1 with the following difference: replace the square plate with an isosceles triangle of width 225 mm at the top and height of 225 mm.

17. Problem 7
The quarter circular uniform gate AB has a width of 6m. The gate controls the flow of water over the edge B. The gate has total weight of 6800 kg and is hinged about its upper edge A. Find P required to keep the gate closed.

18. Problem 8
The quarter circular uniform gate AB has a width of 6m. The gate controls the flow of water over the edge B. The gate has total weight of 6800 kg and is hinged about its upper edge A. Find P required to keep the gate closed.

19. Arch Dam at Idduki