Consider perfect supports at all the edges of the elliptical paraboloid roof given in the previous problem. The spans are 2a = 2b = 20 m, the height of the parabolas are fa = fb = 1.0 m. Determine the membrane forces from snow load, s = 1.5 kN/m2 and the loads on the supports.
Solve Problem
Normal force at point A, Nx [kN/m]= Normal force at point A, Ny [kN/m]= Shear force at point A, Nxy [kN/m]= Normal force at point B, Nx [kN/m]= Normal force at point B, Ny [kN/m]= Shear force at point B, Nxy [kN/m]=Solve
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Steps
Step 1. Write the function of the elliptic paraboloid. Step 2. Give the partial derivatives of the surface function. Step 3. Determine the projected normal forces. The shell structure carries the load as perpendicular arches along x and y axes. where The ratio of the two load components of the undetermined structure can be determined from compatibility condition (assuming identical deflections at the middle). Because of the symmetry of the structure, here Thus the projected normal forces become: Step 4. Calculate the membrane forces at the given points. The membrane forces can be calculated from the projected forces as: The values of the non zero normal forces, Nx and Ny at points A and B are Step 7. Determine the loads on the supports, draw the free body diagram. All of the perfect supports are subjected to tangential forces.Step by step
Check function
Check partial derivatives
Check projected normal forces
Check membrane forces
Step 5. Draw the internal force diagrams.
Check internal force diagrams
Check free body diagram
Results
First the function of the elliptic paraboloid is written. The partial derivatives of the surface function are: The shell structure carries the load as perpendicular arches along x and y axes. The projected normal forces are: where The ratio of the two load components of the undetermined structure can be determined from compatibility condition (assuming identical deflections at the middle). Because of the symmetry of the structure, here Thus the projected normal forces become: The membrane forces can be calculated from the projected forces as: The values of the non zero normal forces, Nx and Ny at points A and B are The free body diagram shows that all of the perfect supports are subjected to tangential forces.Worked out solution
The internal force diagrams are given in the Figure.