File Lecture12.pdf - Lecture 121 Outline: Energy methods...

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Lecture 12 1 Lecture 12 1 Outline: Energy methods Conservation of Energy Castigliano’s theorem Principle of Virtual work 1. Conservation of Energy If a load is applied to an elastic body slowly, then the body deforms and the external work done by the load on the associated deflection e U is transformed to the stain energy stored by the body i U due to the conservation of the energy principle e i U U = . Therefore, upon removal of the external loads the body restores its original position. Conservation of the energy principle can be conveniently used to determine deflections at the point of the load and when only a singular load is acting. Conservation of Energy for trusses Consider a truss subjected to load P applied gradually as shown in Fig .1. Due to the applied load the external work done on the deflection Δ can be written as U e = 0.5 P Δ (see lecture 11 notes). Fig. 1. Truss under gradually applied load P and deflection at the load point Δ Assume that P develops axial force j N in member j . The strain energy stored in member j depends on its length j L , cross-sectional area j A , modulus of elasticity j E and the axial force j N and can be written as (see Lecture 11 notes) 2 2 j j j j j N L U A E = Summing the strain energies for all members of the truss, we can write the total strain energy stored by the truss as 2 1 1 2 2 n j j j j j N L P A E = Δ = (note that there are nine members in the truss so that n =9) When only a single external force is acting on the truss and required displacement is at the load application point and is in the same direction as the force, we can use conservation of energy to find the deflection Δ by using the equation above. 1 Mechanics of Solids Dr Emre Erkmen
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Lecture 12 2 Example The three-bar truss is subjected to a horizontal force of 20kN. If cross-sectional area of each member is 100mm 2 , determine the horizontal displacement at point B , ( ) B h Δ . The modulus of elasticity E = 200GPa. B ) h Since only a single external force is acting on the truss and required displacement is at the same point and in the same direction as the force, we can use the conservation of energy principle. Also note that the support reactions do not produce work because they are not displaced. By using the method of joints, force in each member can be determined as shown on free-body diagrams of pins at B and C . Summation of the strain energies for all members of the truss can be determined as ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 3 1 2 2 2 3 3 3 3 1 2 2 20 10 N 1.732 m 11.547 10 N 1 m 23.094 10 N 2 m 1 20 10 N 2 2 2 2 i i B h i B h N L P AE AE AE AE = Δ = × × × Δ = + + From the above equation, the horizontal deflection at point B can be determined as ( ) 94640.0 N m B h AE Δ = Substituting in numerical values for A and E , we obtain ( ) ( )( ) ( ) 2 2 9 2 3 94640.0 N m 100 mm 1 m /1000 mm 200 10 N/mm 4.73 10 m 4.73 mm B h Δ = = × =
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Lecture 12 3 Example
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File Lecture12.pdf - Lecture 121 Outline: Energy methods...

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