Contact During the Exam

Contact during the exam:

Professor Carl Martin Larsen 95554

Professor Jørgen Amdahl957 45 663

EXAM IN SUBJECT TMR4205BUCLING AND COLLAPSE OF MARINE STRUCTURES

Thursday 29. May 2008

Time: 09.00 – 13.00

Approved help (D):Neither printed nor handwritten notes are permitted.

Approved, simple calculator is permitted.

Results available:22June 2008

The problemtext is on 10 pages and includes 3 problems.

PROBLEM 1

Figure 1

a)For the two the I-profile and the stiffened plate cross-section shown in Figure 1 calculate plastic bending moment, the plastic shear force and the plastic axial force. The yield stress isfy = 300 MPa.

b)For cross-section (b) sketch the stress and strain distribution in bending

- at first yield

- in the elasto-plastic range

- at full plastic utilization

For cross-section (b)indicate both the theoretical position of the neutral axis in pure plastic bending and the position used for practical calculations. Cross-section (b) is first subjected to a plastic bending moment.Then an axial force of N = 2 MN is applied, with the resultant acting in the plate flange. Does this affect the plastic bending moment? Explain your answer.

Figure 2

Figure 3

Figure 2 shows a portal frame. The columns have plastic bending moment Mp1 and the beam has plastic bending moment Mp2.. The frame is subjected to a horizontal load P1 and a vertical load P2. Plastic analysis of the portal frame is performed considering 4 different mechanisms, denoted A,B,C and D.

The results of the analysis with the kinematic(mechanism) method can be presented in terms of failure lines relating critical combinations of the two loads, P1 and P2with respect to formation of the actual mechanism. This is shown in Figure 3 for the case that Mp2 = 2 Mp1 = 2 Mp. The forces are normalized versus Mp/.

c)Use the kinematic method to show that Mechanism D yields failure line 3 (fail 3), given by P1+P2 = 7 Mp/.

d)Relate mechanisms A,B and C to the remaining failure lines. No calculations are necessary. Explain your answers.

e)Show that mechanism D yields the true collapse load when

P1 =2 Mp/, P2 =5 Mp/

Bonus question:

f)For P1 =3 Mp/,P2 =4 Mp/ mechanism D does not give the true collapse load. Can you from inspection of the bending moment diagram and global equilibrium determine the value of P2 ( < 4 Mp/ ) which gives the true collapse load? Sketch the relevant mechanism for this case.

PROBLEM 2

Figure 4 shows a plastic interaction diagram for axial force (N) and bending moment (M) for a circular pipe cross-section. (Note: Axial force may be compressive or tensile). In the same diagram five different histories of axial force and bending moment are plotted. They are obtained from analysis of single structural member with a nonlinear finite element program (usfos). Load versus deformation relationships (P-) obtained in theses analyses are also shown in Figure 4. (The curves are not necessarily drawn to scale)

a)What do we mean with a plastic interaction function? Sketch the stress distribution over the cross-section for states (i) through (iv) on the interaction curve.

b)Relate the axial force-bending moment histories (A….E) to the corresponding load-deformation curves (1….5). Describe the type of load and displacement the structural component has experienced and – when relevant - the boundary condition assumed.

c)Explain specifically what effect that takes place at the points marked by in load-deformation curves 1,3 and 4. Sometimes this is an undesired effect. Explain how the effect can be avoided by the design and how this is taken into account in design codes.

d)Figure 5 shows two models (a) and (b) of a platform. The models are used in nonlinear finite element analysis of the resistance to extreme waves (pushover analysis). The models show the platform prior to application of functional and environmental loads. Displacements for both platforms are magnified by a factor of 40. The resistance from pushover analysis with the two models differs by approximately 10%. From this information, give a physical explanation of the likely cause of this difference. In which of the two analysis results will you have the greatest confidence?

Figure 4

(a) (b)

Figure 5

PROBLEM 3

Figure 6

Figure 6 shows a stiffened panel. The distance between the longitudinal stiffeners/plate width is 0.8 m, and the distance between the transverse girders is 3.2 m. The plate thickness is 16 mm. The plate is subjected to uniformly distributed, combined axial stresses in x-direction and y-direction. The stress in y-direction is y = 70 MPa. The yield stress is F = 360 MPa and the elastic modulus is E = 2.1 105 MPa

a)Calculate the critical compressive stress for the plate according to linear theory when it is loaded uni-axially in x-direction and y-direction, respectively.Sketch the associated buckling modes

b)The plate may be subjected to simultaneous hydrostatic pressure, uniformly distributed. Discuss how this may affect the critical compressive stress for loading in x-direction and y-direction, respectively.

c)The plate will have reserve capacity after buckling. Give a short explanation of the source to this reserve capacity and describe especially the assumptionsused for the boundary conditions of the plate. Sketch qualitatively the average stress -average strain relationship for the plate for compression in x-direction. Sketch also the stress distribution over the plate width and transverse to the long edge at the point of buckling and in the post-buckling range

d)When does the plate reach its ultimate strength? How and where in the plate is failure triggered? Define the concept ‘effective width’

e)Calculate the effective width for compressive load in x-direction. In the y-direction the effective width is 34%. What is the resulting effective width for bi-axial compression?

Provided text book information:

Effective width:

Interaction equation for ultimate compressive strength of plates subjected to bi-axial compression

Buckling Coefficients for Various Load Conditions.