Exercise 3

NTNUNorwegian University of Science and Technology.

Faculty of Marine Technology

Department of Marine Structures. TMR4205 Buckling and Collapse of StructuresBuckling of column - USFOS analysis

Plastic analysis of beams ________________________________________________________________________________________________Date: Jan 2011 Signature: EKim Distributed 29.01.11 Due Date: 5.02.11PROBLEM 1

The Effect of Equivalent Imperfections in Buckling AnalysisThe figure below shows a simply supported column subjected to an axial compressive force, N. The column has a sinusoidal imperfection with maximum amplitude of wio. The critical load, with respect to buckling, of the column is to be determined by using the non-linear computer program USFOS. In this type of analysis, the reference axial load, Nref, is increased step-wise until buckling takes place and the load carrying capacity starts to drop.

a) Calculate the critical loads (cr/Y) for three different initial imperfections and three different reduced slenderness, (a total of nine analyses), so that you will be able to fill in the table below. (Hint: The reduced slenderness can be taken into account by varying either the tube diameter or the member length- use of spreadsheet trial and error calculations recommended).

The input files may be beneficially generated by editing the head and model files given in folder Program Files\USFOS\examples\column Use the GIMPER and GELIMP input statements to generate the imperfection in your input file (both ends rotated). Use the axial displacement at the right end as your control node in the CNODE input line.Imperfection Reduced Slenderness,

0.60.82.6

0.001

0.003(cr/Y)

0.005

b) Plot the peak axial stress from part (a) in a diagram together with (and similar to) the column curves given by ECCS - and ISO19902/Norsok N-004 for tubular beams. For each reduced slenderness, estimate the initial imperfection amplitude, wio, that yields a critical load equal to the one obtained from the column curve given by by ECCS - and ISO19902/Norsok N-004. (In both codes account should be made for the potential influence of local buckling, i.e. a reduced yield stress. Use the ISO formulation also for the ECCS-curve).c) For a reduced slenderness of 0.8 perform analyses with a D/t-ratio of 80 and 30, in addition to that run in pt b. Compare buckling stress with ISO19902/Norsok N-004-predictions. Comment upon the effect of wall slenderness. How does the effect of local buckling manifest itself in the simulations with Usfos?PROBLEM 2

Figure 1 I-beam subjected to uniform load (dim. in mm)

a) Figure 1 shows the cross-section of a beam with I-profile. The moment of inertia is I = 9.2107 mm4. Calculate the plastic section modulus and the shape factor for the profile. What is the plastic bending moment, MP? What is the plastic bending moment, MPred, if the contribution from the web is neglected? Yield stress Y = 300 MPa.

b) What is the plastic shear capacity, QP, of the cross-section? Assume that the cross-section is subjected to combined bending moment and shear force and is in a fully plastic utilized state. Sketch the axial stress distribution and shear stress distribution over the cross-section when the shear force has reached its fully plastic capacity Q = QP. Sketch qualitatively the bending moment shear force capacity interaction function for the cross-section (no calculations!) Use relative magnitudes M/MP and Q/QP.

c) The beam is subjected to a uniformly distributed load, q, as shown in Figure 1. The ends of the beam can be assumed to be rotationally clamped. Introduce a reasonable collapse mechanism and calculate the plastic collapse load for the beam. Comment your choice of bending moment values in the plastic hinges, and check afterwards if your choice was reasonable.

d) Compared to the load causing first yield in the beam there are two sources of additional resistance, which is accounted for in the plastic collapse load. What are they? Sketch qualitatively and explain (key words only!) the load versus mid span deformation curve for the beam up to the plastic collapse load level. For the calculation of the collapse load to be correct, the cross-section of the beam must satisfy certain compactness requirements. What do we mean by this?

PROBLEM 3 (Voluntary)A beam with axially fixed ends and clamped at support A is subjected to a concentrated load P at mid span as shown in Fig. a.

The beam shall be analysed by means of plastic theory for finite displacements. The assumed mechanism is shown in Fig. b.

a) Show by means of geometric considerations that the total virtual elongation, , of the beam can be expressed by the virtual rotation, , as:

The virtual elongation is further split to the yield hinges proportional to the virtual rotation of the hinges.

b) Use the virtual work principle to show that the equilibrium equation can be written as

STYLEREF\ \* MERGEFORMAT

STYLEREF1\n \* MERGEFORMAT

where M and N are bending moment and axial force in yield hinges.

c) It is assumed that the following interaction function applies for the cross-section

where MP and NP are plastic bending moment and plastic axial force. Determine the coefficients C1 and C2 STYLEREF\ \* MERGEFORMAT

STYLEREF1\n \* MERGEFORMAT .

Given:

STYLEREF\ \* MERGEFORMAT

STYLEREF1\n \* MERGEFORMAT d) What do the consistency criterion and the normality criterion express? Use the normality criterion and use the kinematic condition from a) to show that the axial force can be related to the lateral displacement by:

e) Show that the equilibrium equation can be expressed as

where

What is the validity range of the expression. What does PO represent?

f) Derive the equilibrium equation for the pure membrane effect by means of a static consideration. Check the continuity of the load-deformation curve in the transition form combined bending - axial force to pure membrane solution.

15

10

200

250

15

4000

q

A

P

B

/2

/2

A

w

B

/2

/2

1

_1074078025.unknown

_1074430774.unknown

_1074430886.unknown

_1074431013.unknown

_1074430796.unknown

_1074078501.unknown

_1074430742.unknown

_1074079195.unknown

_1074078333.unknown

_937146060.unknown

_937146204.unknown

_937143223.doc

z

y

x

N

N

A

A

L = 10m

EMBED Equation.2

Section A-A

5mm

240mm