A parabolic barrel vault with the same geometry given in the previous problem is subjected to self-weigth. The supports are different: one arch is free, the other arch and the straight edges are perfect supports. The thickness of the reinforced concrete shell is h = 80 mm. The weight density is 25 kN/m3. Determine the membrane forces, and the loads on the supports, draw the free body diagram.
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
Do you need help?
Steps
Step 1. Calculate the self-weight load. Step 2. Write the function of the parabolic surface. Step 3. Give the partial derivatives of the surface function. Step 4. Determine the projected normal forces. Step 5. Calculate the membrane forces at the given points. The membrane forces can be calculated from the projected forces as: The values of the meridian forces at points A and B are The barrel vault carries a part of the load as a cantenary arch, while the remaining part analogously to a cantilever beam. Step 7. Determine the loads on the supports, draw the free body diagram.Step by step
Check load
Check function
Check partial derivatives
Check projected normal forces
Check membrane forces
Step 6. Draw the internal force diagrams.
Check internal force diagrams
Check free body diagram
Results
First the self-weight of the structure is calculated: The function of the parabolic surface is written in the following form: The partial derivatives of the surface function are: Then the projected normal forces are determined. The values of the meridian forces at points A and B are The barrel vault carries a part of the load as a cantenary arch, while the remaining part analogously to a cantilever beam. The internal force diagrams are given in the Figure. The free body diagram and the loads on the supports are shown below.Worked out solution
The membrane forces can be calculated from the projected forces as: