Engineering Mechanics Equivalent force systems: problems

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.

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

Special Force Systems Parallel force system Coplanar force system

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.

Tug Boats Tugboats are used to tow big liners etc. to the shore. http://en.wikipedia.org/wiki/Tugboat

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

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

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

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

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

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.

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

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

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

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.

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.

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.

Arch Dam at Idduki