NTNU

NorwegianUniversity of Science

and Technology.

Faculty of Marine Technology

Department of Marine Structures.

Exercise3

TMR4205 Buckling and Collapse Structures

Plastic analysis of frame

______

Date: January 2012 Signature: EKim Distributed Date: 23.01.12 Due Date:30.01.12

Figure 1 shows a portal frame subjected to a horizontal load at the upper left corner,

and a vertical load at the mid-span of the beam. All members are of pipe cross-

sectional shape having a dimension of 500mmx20mm. The yield stress is 300MPa

and 360MPa for columns and beam, respectively, and the bending moment capacities

are Mp and 1.2Mp for columns and beam, respectively. L is given in meters.

V

H

1.2 Mp

Mp

0.7L

0.7L

Mp

L

Figure 1. A portal frame.

a) Sketch the potential collapse mechanisms and calculate the associated plastic

collapse loads.

b) Present the solutions from part (a) in an x-y plot where thex-axis represents the

horizontal load, and they-axis represent the vertical load. You will observe that

the graph splits the area into various domains, explain what do the domains

represent? What collapse mechanism is governing when V2HPand

VHP , respectively.

c)For V2 H=P, calculate the elastic moment distribution by means of the Unit Load method or by using the following information: At the upper right corner of the frame (Figure 1) the elastic bending moment is equal 0.37PL. (Hint: In the present case, it is convenient to introduce a hinge at theupper right corner, in order to make the structure statically determinate).

d) Determine the load level corresponding to the first plastic hinge and calculate the

corresponding horizontal deformation of the frame. Express the load and

deformation in terms of the plastic moment capacity, Mp. (Hint: Apply a unit load

just after the plastic hinge has been created).

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e) Introduce a plastic hinge at the critical section and calculate the bending moment

distribution for additional loading. Determine how much additional loading the

frame can take before the second hinge is created. What is the total load at this

stage. Compare the results with those of the mechanism analysis.

Given: The integration of two moment diagrams.

a

c

M



∫

M M

dx

2

0 7 L

5

1

−

ab

1

c

  • L


0.7L

b

0.7L

M1

1 4 L

EI

12

2

12



f)Use the computer program USFOS to calculate the load as a function of horizontal

displacement. Assume initially thatV2 HP and increase the load step-wise

until a plastic mechanism has been reached. Scale the applied load such that the

reference load, Pref, (to be given in the input line NODELOAD) is 0.10 times the

calculated critical load. In this way the expected collapse load factor should be 10,

i.e. Pcr = 10 x Pref. Indicate, specifically, when the hinges are formed. (L = 12m).

g) Compare the results of the analysis in part (f) with those of analytical calculations.

What are the major reasons for discrepancies between analytical and numerical

results. Explain what happens when the external loads are removed (unloaded). Is

the frame stress free?

h) Repeat USFOS analysis for VHP , andH0 : VP . How is the sequence

of plastic hinge formation in these cases?

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