Structural Mechanics Assignment Help
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Energy Methods in Elasticity Problem 1: Consider an elastic cantilever beam loaded at its tip
a) Specify the boundary conditions. b) Derive the load-tip displacement relation using four methods presented in class. Method I Solving uncoupled problems Method II Solving coupled problem (direct integration) Method III Castigliano Theorem Method IV Ritz Method
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Solution 1: (a) Boundary conditions w(0) = 0 w’(0) = 0 (b) Derive the load-tip displacement relation using four methods 1. Solving second order uncoupled problems We will use moment-curvature relationship curvature-displacement relationship to solve the problem. These relationships combine to give: We need the moment. Find reactions
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Find moment in beam base on force balance and moment balance
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Combing the above results from force balance and moment balance M = -Pl + Px Integrate moment once
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Use B.C.
Integrate once
Use B.C.
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2. Solve coupled problems using 4th order direct integration We use the 4th order differentiated equation:
And we need four boundary conditions:
Integrate and use boundary conditions to determine the constants
Use B.C.
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We get
Then
Use B.C.
We get
Then
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Use B.C.
We get Finally
3. Castigliano Theorem Find moment in beam
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Calculate strain energy
Use Castigliano Theorem
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4. Ritz method Assume shape function
Use boundary conditions
We get
Also
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Find total potential energy
Minimize3
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Solve for C1 and C2 using the above two equations
Finally
Note: we obtain the exact solution with
If we were to assume shape function of lower order, we can only obtain an approximate solution. solidworksassignmenthelp.com
Problem 2: Consider a beam of length L and bending rigidity EI which is fully clamped on both ends shown in Figure 1. The beam is subjected to a point force P at the midspan. Solve the problem (find the expression between the load and the deflection under the load) using:
a) The direct integration of beam equation with the suitable boundary conditions (exact solution). b) The Ritz Method (approximate solution). c) Compare the results and calculate the relative error of the Ritz Method.
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Solution:2
Solve for half the beam. Boundary conditions are:
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a) Direct integration We use the 4th order differentiated equation:
where q = 0 Integrate once Use B.C. (4), we have
By integration
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Use B.C. (2), we have
Use B.C. (3),
Integrate once
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Use B.C. (1),
Mid-span deflection
b) Ritz method
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Assume shape function
Boundary conditions are:
Use B.C. (5) and (6),
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Apply B.C. (7) and (8),
Combine the above two equations, we can solve for
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Calculate U = W
Minimize
Finally
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c) Compare Ritz method and exact solution
Because we guess the correct order of the shape function, we got the exact solution. %error=0 Problem 3: Use Castigliano’s Theorem to calculate the horizontal deflection at point D in Figure
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Solution 3: Find reaction
where and Rx MR are reaction force and reaction moment Find internal moment distribution at each section
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Section AB Section BC
Section CD
Calculate strain energy
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Use Castigliano
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