How long does it take for the temperature at x= 2 to reach T= 80C?Calculate

Q1.Consider a ten-story building as a lumped mass model. Masses M1 … M10 are ‘lumped’ at ‘nodes’ 1 to 10Stiffness K1 … K10 prevent relative lateral movement (drift)Forces F1 … F10 are applied at nodes 1 to 10Assume that all the values of M, K and F are 1 unit. Set equations of equilibrium by simple finite elements and calculate the displacement using both Gauss Seidel iteration and Gaussian Elimination. Compare and discuss your answers for the above two methods.[33marks]
2020/21CIVI0008Dr. K. F. Tee Page 2of 3

Q2.The partial differential equation for one-dimensional transient state heat flow is22xTKtT=A metal bar is 10 cm long and has K= 1 cm2/second. Initially, the temperature T of the bar is a uniform 0C. At time t= 0 one end of the bar at x= 0 is brought into contact with a heat sink with a temperature of 100C while the other end is brought into contact with a heat sink with a temperature of 50C.

Solve the temperature distribution of the bar from t= 0 until t = 20 seconds. Plot the temperature distribution vs. time for different parts of the bar (x= 2, 4, 6 and 8) and then answer the following questions.

a)How long does it take for the temperature at x= 2 to reach T= 80C?

b)How long does it take for the temperature at x= 4 to reach T= 60C?

c)How long does it take for the temperature at x= 6 toreach T= 50C?

d)How long does it take for the temperature at x= 8 to reach T= 50C?[33marks]
2020/21CIVI0008Dr. K. F. TeePage 3of 3

Q3.Consider an anti-symmetric vibration pattern (mode shape) of the suspension bridge. The partial differential equation (PDE) governing vertical deflections vin this mode is:442222xvBxvAtv−=where A= 500 m2s-2(due to gravity stiffness of the cable)andB = 2 x 105m4s-2(due to flexural rigidity of the deck) Figure 1shows an idealisation of the bridge. The span Lbetween the supports at nodes -2 and 2 is taken as 40 m.The deflections are anti-symmetric about the midspan x= 0. The boundary conditions are:v= 0, 022=xvat x= -20, 20 (nodes -2 and 2)Initially the bridge is at rest (zero velocity) and has the deflected shape given by )/2sin(1.00Lxv=m. The bridge isreleased and oscillates.

a)Write down the finite difference form of the PDE for position iat time j.

b)Using the given values of A and B, a time step of t= 0.1 second and x=10 m, draw the PDE template to be appliedat each node.

c)Use the boundary conditions and anti-symmetry to show that the template need only be applied at one location e.g. i= -1 for each time step j. Hence draw the resulting simplified template at location i= -1.

d)Apply the initial condition to obtain the deflected shape after one time step.

e)Obtain an approximate value for the time taken by the bridge to complete one cycle of oscillation and return to the original position at time t= 0.-3 -2 -1 0 1 2 3Figure 1Idealisation of bridge