The cross section of a water tank is given in the Figure. The top of the walls is connected by a floor which hinders displacement and allows rotation of the edge of the wall. Stiffness of the base slab is: Ds = 11 600 kNm2/m, stiffness of the wall is Dw = 9 300 kNm2/m. The soil is modelled as an elastic (Winkler type) support, the foundation coefficient is: c = 100 000 kN/m3. The tank is fully filled with water.
a) Determine the moments, Mx in the walls and in the slab, calculate the tensile force in the floor. (Consider only a strip of unit width far from the end walls. The strip is long, λL > 4.)
b) Determine the internal forces caused by 30 degrees heating of the floor. (Elastic compressibility of the floor is neglected.) Thermal expansion coefficients of steel and concrete are the same: α = 1.2×10-5 1/°C.
c) How the moments due to the water pressure will change if the connection of the floor and the wall fails?
Solve Problem
Problem a) Moment at the bottom of the wall, M [kNm/m]= Tensile force in the floor, N [kN/m]= Problem b) Tensile force in the floor, N [kN/m]= Problem c) Moment at the bottom of the wall, M [kNm/m]=Solve
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Steps
Problem a) Solve the statically indetermined structure with the force method. The primary structure is obtained by introducing a hinge at the intersection of the wall and the slab. The corresponding deflected shape and bending moment curve are shown in the Figure, the relative rotation in the hinge (simply supported beam subjected to a triangular load) is Step 2. Draw moment diagram of the primary structure from the redundant. Calculate the displacement at the removed support. The redundant is a moment couple, the deflections and moments are shown in Figure. The relative rotation contains two terms, the first one is the end rotation of the wall (simply supported beam subjected to an end moment), while the second one is the end rotation of the slab on elastic foundation is given below. Since and , the slab is considered to be long, and we write Step 3. Write compatibility condition. Determine the redundant. Give the internal forces of the statically indetermined structure. The compatibility condition is: The bending moment is obtained by superposition: Tensile force in the top support (floor) is Problem b) Solve the statically indetermined structure with the force method. Top displacement is the half of the elongation of the heated top floor: (From the water load there is no displacement on top, see Problem a) ) Step 2. Give the top displacement from the redundant. Step 3. Write compatibility condition. Determine the redundant. Give the internal forces of the statically indetermined structure. The compatibility condition is: Problem c) Step 1. Calculate the moment of the statically determined structure. The moment is determined on a cantilever:Step by step
Show moment diagram from the load
Show moment diagram from the redundant
Show total moment diagram
Step 1. Define the primary structure. Calculate the displacement at the removed support from the temperature load.
Show displacement from temperature change
The primary structure is a cantilever.
Show displacement from the redundant
Show total moment diagram
Show moment
Results
Problem a) The statically indetermined structure is solved by the force method. The redundant is a moment couple, the deflections and moments are shown in Figure. The relative rotation contains two terms, the first one is the end rotation of the wall (simply supported beam subjected to an end moment), while the second one is the end rotation of the slab on elastic foundation is given below. Since and , the slab is considered to be long, and we write The compatibility condition is The bending moment is obtained by superposition: Tensile force in the top support (floor) is Problem b) The statically indetermined structure is solved by the force method. Top displacement is the half of the elongation of the heated top floor: (From the water load there is no displacement on top, see Problem a) ) The top displacement from the redundant: The compatibility condition is Problem c) The moment is determined on a cantilever:Worked out solution
The primary structure is obtained by introducing a hinge at the intersection of the wall and the slab. The corresponding deflected shape and bending moment curve are shown in the Figure, the relative rotation in the hinge (simply supported beam subjected to a triangular load) is
The primary structure is a cantilever.