Norwegian University of Science and Technology

Exercise

Solution

NTNU

Norwegian University of Science and Technology.

Faculty of Marine Technology

Department of Marine Structures.

SIN 1048 Buckling and Collapse of Structures

Shape factor and plastic analysis of beam

______Date:January 2003 Signature: JAm Distributed Date: Due Date:

Problem One

a) The stress distribution over the cross-section is sketched in Figure 1. Since this is an unsymmetrical cross-section, it is important to note that the position of the neutral axis in a fully plastic state will not be the same as at first yield state. The neutral axis will move towards the side of smaller stress value.

Figure 1. Stress Distributions.

A plastic hinge can be introduced when a fully plastic state has been reached. This is simply because at that point the section can not carry any more moment.

b) I order to obtain the elastic and plastic section modulus, we need to calculate the neutral axis for the elastic and fully plastic state. With reference to Figure 2, the neutral axis at the elastic state can be calculated as follows.

On the other hand, the neutral axis at fully plastic state, is calculated such that the area between the two sides of the cross-section (compression and tension sides) are equal. Therefore,

Figure 2. The Neutral Axis at Elastic and Plastic States.

The elastic section modulus is given by,

(1)

where Inn is the moment of inertia about the neutral axis, and y is the longest distance perpendicular to the neutral axis which in this case is obviously to be towards the upper flange. By using Figure 2, we obtain,

By definition, the plastic section modulus is given by,

(2)

where Y is the yield stress and Mp is the plastic moment. From Figure 3, we can write

(3)

where,

Therefore,

Figure 3. Plastic Section Modulus.

c) The shape factor is given by,

This is a higher value as compared to those of typical I-profiles given in the Søreide’s book, (i.e 1.1 It can be inferred that the unsymmetrical I-profiles have higher shape factors compared to symmetric ones. Considering the given profile in this exercise, the factor will decrease with increasing size of the top flange until symmetry has been reached.

d) The shape factor of a cross-section represents the reserve strength of that section after first yield.

Problem Two

a) There are three possible collapse mechanisms for the given structure as shown in Figure 4. The associated collapse loads can be calculated from the principle of virtual work, Equation (4).

(4)

Figure 4. Possible Collapse Mechanism.

(i) Mechanism I:

(ii) Mechanism II:

(iii) Mechanism III:

b) From the above results, it is natural to select the true collapse load from mechanisms I and II as,

Drawing the moment diagram will prove that this is the true collapse load if the moment values do not exceed the plastic moment capacity on any span. That is,

Consider mechanism I and assume that the moment diagram has the shape shown in Figure 5. Then,

The moment equilibrium of the left span (about point A) yields,

and the moment equilibrium of span BC (about point B) yields,

Static equilibrium is satisfied by

Figure 5. Moment Diagram.

1