Structural Analysis

Q1 Calculate the plastic modulus Sx of the section shown below. All dimensions are in mm. Note: The position of the plastic neutral axis can be found from the consideration that the cross- sectional areas in compression and tension are equal. 200 X IY 80120 180 120|80| X 260 100 150

Q1: Determine all reaction components of the beam shown in Figures 1, using the Force Method Draw the Shear and Bending Moment diagrams

Q1: The crane travels along a runway girder that is supported on columns at B, C and E as shown in Figure 1. Draw the Influence Lines with the main values for: a) vertical reaction at D b) shear at C c) moment at E; you may use Multiframe to assess the main values

1. Fracture of a thick-walled pressure vessel (10 points) A thick-walled pressure vessel has inner radius >= 10 mm and outer radius R = 40 mm. A flaw, depth a = 1 mm, is detected on the internal surface of the vessel. The vessel is made of a steel with a yield strength or= 200 MPa and a fracture energy 10³ J/m². The Young's modulus of the steel is 200 GPa. Estimate the pressure that the vessel can sustain. Hint: The stress intensity factor for an internal crack in a thick-walled pressure vessel can be estimated by the following expression (Underwood, J. H. (1972). Stress intensity factors for internally pressurized thick-wall cylinders. ASTM International):

2. Mode mixity of the four-point bend specimen (10 points) a) Show that the energy release rate for the four-point bend specimen isA where M is the applied moment per unit thickness, h and H are the thickness of the two layers. Assume that both layers have the same Young's modulus E. Hint: The following two papers developed the technique. P.G. Charalambides, J. Lund, A.G. Evans, and R.M. McMeeking. A test specimen for determining the fracture resistance of bimaterial interfaces. Journal of Applied Mechanics. 56, 77-82 (1989). R.H. Dauskardt, M. Lane, Q. Ma, N. Krishna. Adhesion and debonding of multi-layer thin film structures. Engineering Fracture mechanics 61, 141-162 (1998). b) Charalambides et al (1989) used the finite element method to determine the mode angle. When the two layers have identical elastic constants and identical thickness, one can determine the mode angle analytically. Show that in this special case

3. Fatigue of a pressure vessel (10 points) A cylindrical steel pressure vessel of 7.5 m diameter and 40 mm wall thickness is to operate at working pressure of 5.1 MPa. The fracture toughness for the steel is 200 MPa√m. The growth of the crack by fatigue may be represented approximately by the equation where A =2.44×10¹ (MPa) m¹. The design assumes that failure will take place by fast fracture from a crack which has extended gradually along the length of the vessel by fatigue. To prevent fast fracture, the total number of loading cycles from zero to full load and back to zero again must not exceed 3000. Find the minimum pressure to which the vessel must be tested before use to guarantee against failure in under the 3000 load cycles. Hint: The stress intensity factor for a crack along the length of a thin-walled pressure vessel can be estimated by the following expression (Tada, H., Paris, P. C., & Irwin, G. R. (1973). The stress analysis of cracks. Handbook, Del Research Corporation, 34, 635): K₁ = √na-F(2),

Problem 3: 9-23. Determine the displacement at point D. Use the principle of virtual work. El is constant.

Write a MATLAB program for the analysis of a 2D truss using matrix method analysis and for showing the deformation of truss through animation.

Analyze the 3D steel trusses shown in Figs. 3 and 4 using the MATLAB program. The truss is supported by ball and socket joints. Verify all intermediate calculations manually for the truss of Fig. 3. E = 2x105 MPa.

Problem 1: 9-15. Determine the vertical displacement of joint A. Assume the members are pin connected at their end points. Take A=3in² and E=29(10³)ksi for each member. Use the method of virtual work.