Determine the strain energy of the cantilever loaded by a concentrated force, F = 5 kN at the endpoint. Length of the beam is L = 6 m, bending stiffness of the cross section is EI = 4.2 × 106 Nm2.
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Strain energy, U [Nm]=Solve
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Steps
Hint: integral of two functions: , when at least one of the functions is linear can be performed by the so called “visual integration method”. Let f2 be linear. where is the integral of function f1, i.e. the area under the function on the 0 – L interval, while h is the ordinate of f2 under the center of gravity of A. Step 1. Draw the moment diagram. Give the functions of the moment and the curvature along the beam’s length. Step 2. Determine the strain energy of the bent beam by performing the integration. The strain energy is Step 3. Determine the strain energy of the bent beam applying the visual integration method. Since is linear, the result of the above integration can be obtained by multiplying the area of the moment diagram with the value of the linear curvature function at the centroid of the moment diagram:Step by step
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Results
Hint: integral of two functions: , when at least one of the functions is linear can be performed by the so called “visual integration method”. Let f2 be linear. where is the integral of function f1, i.e. the area under the function on the 0 – L interval, while h is the ordinate of f2 under the center of gravity of A. The functions of the moment and the curvature along the beam’s length are given in the Figure below. The strain energy of the bent beam is Since is linear, integration can be performed by the visual integration method by multiplying the area of the moment diagram with the value of the linear curvature function at the centroid of the moment diagram:Worked out solution