Determine the bending moment from edge disturbance at the curved edge of the barrel vault of Problem 11.6. (The parabolic barrel vault is subjected to snow load, s = 1.5 kN/m2. It is supported by arches at both curved ends and by beams at the straight edges. The width of the structure is l = 12 m, the length is b = 10 m and the height is f0 = 3.5 m.) Thickness of the structure is 8 cm.
Solve Problem
Maximum bending moment assuming hinged edges, Mmax [Nm/m]= Maximum bending moment assuming built-in edge, Mmax [Nm/m]=Solve
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Steps
The only non zero membrane force is The values of the normal force, Ny at points A and B are Step 2. Determine maximum moment from the edge disturbance. The membrane force, Ny of the barrel vault results in displacements of the edges which are hindered by the supporting arches. According to Geckeler’s approximation the bending moment at the support is determined by fitting an osculating cylinder to the edge of the vault. Considering hinged support the maximum moment is where the hoop force of the osculating cylinder, is equal to the membrane force parallel to the edge, Ny, which varies along the curved edge. The maximum value of the membrane forces arise along the longitudinal edge , thus the maximum moment is The location of the positive maximum along x axis is from the edge, where the radius, R is the inverse of the curvature of the osculating cylinder: The location of the maximum positive moment is Considering built-in support the maximum negative moment is The maximum moment arises at the corner : Since in many cases the resistance of a support is hard to calculate, we may design the shell for both bending moments.Step by step
Step 1. Give the membrane solution of the barrel vault.
Check membrane solution
Check maximum moment assuming hinged edges
Check maximum moment assuming built-in edges
Results
The only non zero membrane force is The values of the normal force, Ny at points A and B are The membrane force, Ny of the barrel vault results in displacements of the edges which are hindered by the supporting arches. According to Geckeler’s approximation the bending moment at the support is determined by fitting an osculating cylinder to the edge of the vault. Considering hinged support the maximum moment is where the hoop force of the osculating cylinder, is equal to the membrane force parallel to the edge, Ny, which varies along the curved edge. The maximum value of the membrane force arise along the longitudinal edge , thus the maximum moment is The location of the positive maximum along x axis is from the edge, where the radius, R is the inverse of the curvature of the osculating cylinder: The location of the maximum moment is Considering built-in support the maximum negative moment is The maximum moment arises at the corner : Since in many cases the resistance of a support is hard to calculate, we may design the shell for both bending moments.Worked out solution
First the membrane solution of the barrel vault must be derived.