Norges Teknisk Naturvitenskapelige Universitet
Fakultet for Ingeniørvitenskap og Teknologi
Institutt for Marin Teknikk
Contact person:
Name: Jørgen Amdahl
Phone: 95544
MID TERM TEST
TMR 4205
BUCKLING AND COLLAPSE OF STRUCTURES
Date:
Monday 14 March 2005
Time:
0915-1100
Credits:
7.5 Sp
This mid term test counts 25% of the final grade
Allowed during exam:
C – Type-approved, simple calculator is allowed.
No printed of hand-written material
THIS SET CONSISTS OF 3 PAGES
TRM 4205 BUCKLING AND COLLAPSE OF STRUCTURES
Mid-term test 14 March 2005
Page 2 of 3
PROBLEM 1
15
q
10
250
4000
15
200
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 = 9.2·107 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?
TRM 4205 BUCKLING AND COLLAPSE OF STRUCTURES
Mid-term test 14 March 2005
Page 3 of 3
PROBLEM 2
EI
N
w

Figure 2
a) Figure 2 shows a column subjected to an axial compressive force. Sketch
qualitatively (no calculations) and in the same diagram the relationship
between the axial force, N, and the lateral displacement, w, for the column for
the following cases:
i. Column is perfectly straight
ii. For an initially imperfect column, for two different magnitudes of
maximum out-of-straightness.
iii. Perfect column – large deflection solution
What do we mean by a bifurcation point? Show the position of a bifurcation
point in the diagram.
b) When the axial force is zero and the beam-column is subjected to a uniformly
distributed lateral load of 2.5 kN/m, the mid span displacement is 20 mm and
the mid span moment is 5 kNm. Estimate mid span displacement and the mid
span moment when the axial force is N = 5 EI/2 and the lateral load is 5.0
kN/m.
c) Describe how we can develop a column buckling curve using the concept of
an equivalent initial imperfection. What are the major effects the equivalent
imperfection accounts for? Set up the expression for the failure criterion, on
which the column curve is based (Solution for the critical stress is not
required)