Engineering Mechanics. Equivalent force systems: problems
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1 Engineering Mechanics Equivalent force systems: problems
2 A 36-N force is applied to a wrench to tighten a showerhead. Knowing that the centerline of the wrench is parallel to the x axis. Determine the moment of the force about A.
3 The frame ACD is hinged at A and D and is supported by a cable that passes through a ring at B and is attached to hooks at G and H. Knowing that the tension in the cable is 1125 N, determine the moment about the diagonal AD of the force exerted on the frame by portion BG of the cable.
4 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.
5 Moment about a point
6 Couple
7 Equivalent systems
8 Multiple force moment
9 Equivalent Force Moment Systems Couples M j and forces F i 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
10 Special Force Systems Parallel force system Coplanar force system
11 Example 1 Determine the resultant of the four forces and one couple which act on the plate shown.
12 Solution
13 Solution
14 Vector approach
15 Tugboat Problem 1. 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.
16 Tug Boats Tugboats are used to tow big liners etc. to the shore.
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19 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
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22 Moment in 3D
23 Moment about an axis
24 Distance between two lines L and F M = r F M = F 2 d q F 2 = F 2 F1 2 F 1 =(F ) F 2 = F F 1
25 Couple in 3D
26 Force-couple system
27 Multiple forces and moments
28 Simplest Resultant
29 Wrench
30 Solved problem 3.12 BJ 10ed
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34 Wrench
35 Step-1
36 Step-2 Step-3
37 Step-4 Step-5 Step-6 Intersection with x-y plane z = 0 Step-7 Same procedure for arbitrary force-couple
38 Using Symmetry to convert 3D problems to 2D (adapted from 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
39 Problem 3 Find the simplest resultant for the forces acting on the simply supported beam
40 Simple Example (
41 Distributed forces Forces act per unit volume/area/length Per unit volume (N/m 3 ) Density uniformly charged bodies (involving charge density) Per unit area (N/m 2 ) 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
42 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.
43 Hydrostatic Forces Act normal to the surface Force per unit area (N/m 2 ) (pressure) The total pressure at depth z is: p = ρgz = γz
44 Non-Uniform Forces ( Hydrostatic Forces
45 Non-Continuous non-uniform Loading Load replaced by an equivalent triangular loading
46 Problem 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. 2. 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.
47 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.
48 Problem 8 1. 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.
49 Arch Dam at Idduki
50 The gate ABC (cross-section shown) is made of two rectangular panels AB and BC having uniform thickness and welded together at B. The gate is 500 mm wide and is used to maintain water level in two adjacent channels. The gate is pivoted about a horizontal axis through A by hinges located along its top edge. To ensure that the depth of water does not exceed 1500 mm in the left channel, determine the maximum value of the total weight of the gate. The depth of water in the right side channel is 900 mm. Neglect the thickness of the gate in calculating the center of gravity, and the reduction in its weight due to buoyancy effects. Take the specific gravity of water = 10 kn/m 3.
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57 If the combined moment of two forces about point C is zero, determine (a) the magnitude of the force P (b) the magnitude R of the resultant of the two forces (c) the coordinates x and y of the point A on the rim of the wheel about which the combined moment of the two forces is a maximum (d) the combined moment MA of the two forces about A
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60 Tutorial-1: Additional Problems
61 Extra Problem 1 Determine the wrench resultant of the three forces acting on the bracket. Calculate the coordinates of the point P in the x-y plane through which the resultant force of the wrench acts. Also find the magnitude of the couple M of the wrench.
62 Extra Problem 2 Replace the two forces and the negative wrench by a single force R applied at A and the corresponding couple M.
63 The resultant of the two forces and couple may be represented by a wrench. Determine the vector expression for the moment M of the wrench and find the coordinates of the point P in the x-z plane through which the resultant force of the wrench passes.
64 Tutorial-2: Additional Problems
65 Extra Problem 1 W min = 41.8kN A hollow steel cone with internal dimensions as shown in Figure has a pinhole at the top. The cone is filled with water. What is the minimum weight of the cone which will prevent the water from up-lifting the cone and flowing out? Use g = 10 m/s 2.
66 Extra Problem 2 p = 7.48 MPa Oil pressure in the vertical hydraulic cylinder of length 1.5 m shown in Figure controls opening and closing of the vertical rectangular gate against the pressure of fresh water on the opposite right-hand side. The gate has a horizontal width of 2 m perpendicular to the plane of this paper. For a depth h = 3 m of water, calculate the required oil pressure p which acts on the 150-mm- diameter piston of the hydraulic cylinder. Specific weight of fresh water is 10 kn/ m 3.
67 Extra Problem 3 M = 17.3N.m The gate AB is a a 260-kg rectangular plate 1.5 m high and 1.1 m wide and is used to close the discharge channel at the bottom of an oil reservoir. As a result of condensation in the tank, fresh water collects at the bottom of the channel. Calculate the moment M applied about the hinge axis B required to close the gate against the hydrostatic forces of the water and oil. The specific gravity of the oil is Density of water is 1000 kg/m 3 and acceleration due to gravity g = 9.81 m/s 2.
68 Extra Problem 4 A buoy in the form of a uniform 8-m pole 0.2 m in diameter has a mass of 200 kg and is secured at its lower end to the bottom of a fresh-water lake with 5 m of cable. If the depth of the water is 10 m, calculate the angle made by the pole with the horizontal.
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