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Solution

Indeterminate systems are solved using the Clebsch Method in a similar way to displacements in determinate systems, with the difference being that there is one additional equation (boundary condition) and one additional unknown (reaction).

Clebsch Method

Equilibrium equations

\sum 𝑋 = 0 𝑅 2 = 0 \sum 𝑌 = 0 𝑅 1 + 𝑅 3 - 5 𝑞 𝑙 = 0 𝑅 1 = 5 𝑞 𝑙 - 𝑅 3 \sum 𝑀 1 = 0 𝑀 𝑈 - 𝑅 3 * 3 𝑙 + 252 𝑞 𝑙 2 = 0 𝑀 𝑈 = 3 𝑙 𝑅 3 - 252 𝑞 𝑙 2

Classical Clebsch Method

𝑀 𝑔 = 𝑀 𝑢 * 𝑥 0 + 𝑅 1 𝑥 - 12 𝑞 𝑥 2 + 𝑅 3 (𝑥 - 3 𝑙) 𝑀 𝑔 = 3 𝑅 3 𝑙 𝑥 0 - 252 𝑞 𝑙 2 𝑥 0 + 5 𝑞 𝑙 𝑥 - 𝑅 3 𝑥 - 12 𝑞 𝑥 2 + 𝑅 3 (𝑥 - 3 𝑙) 𝐸 𝐼 𝑤 ″ = - 𝑀 𝑔 𝐸 𝐼 𝑤 ″ = - 3 𝑅 3 𝑙 𝑥 0 + 252 𝑞 𝑙 2 𝑥 0 - 5 𝑞 𝑙 𝑥 + 𝑅 3 𝑥 + 12 𝑞 𝑥 2 - 𝑅 3 (𝑥 - 3 𝑙) 𝐸 𝐼 𝑤' = - 3 𝑅 3 𝑙 𝑥 + 252 𝑞 𝑙 2 𝑥 - 5 𝑞 𝑙 𝑥 2 2 + 𝑅 3 𝑥 2 2 + 12 𝑞 𝑥 3 3 - 𝑅 3 (𝑥 - 3 𝑙) 2 2 + 𝐶 𝐸 𝐼 𝑤 = - 3 𝑅 3 𝑙 𝑥 2 2 + 252 𝑞 𝑙 2 𝑥 2 2 - 5 𝑞 𝑙 𝑥 3 6 + 𝑅 3 𝑥 3 6 + 12 𝑞 𝑥 4 12 - 𝑅 3 (𝑥 - 3 𝑙) 3 6 + 𝐶 𝑥 + 𝐷

Boundary conditions

𝑤' (0) = 0 𝐶 = 0 𝑤 (0) = 0 𝐷 = 0 𝑤 (3 𝑙) = 0 - 3 𝑅 3 𝑙 322 𝑙 2 + 252 𝑞 𝑙 2 32 * 𝑙 2 2 - 5 𝑞 𝑙 33 * 𝑙 3 6 + 𝑅 3 33 * 𝑙 3 3 + 12 𝑞 34 * 𝑙 4 12 = 0 - 9 𝑅 3 𝑙 3 + 2978 𝑞 𝑙 4 = 0

Reactions

𝑅 3 = 338 𝑞 𝑙 𝑅 1 = 78 𝑞 𝑙 𝑀 𝑈 = - 18 𝑞 𝑙 2

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Clebsch Method

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Clebsch Method