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References
Bazant, Z.P. and Cedolin, L., 1991, Stability of Structures, Oxford University Press, Oxford.
Chwalla, E., 1959, Hilfstafeln zur Berechnung von Spannungsproblemen der Theorie II. Ordnung
und von Knickproblemen, Stahlbau Verlag, K oln.
Collatz, L., 1960, The Numerical Treatment of Differential Equations, 3rd ed., Springer-Verlag, Berlin.
Leipholz, H., 1968, Stabilitatstheorie, Teubner Verlag, Stuttgart.
Pfluger, A., 1964, Stabilitatsprobleme der Elastostatik, 2nd ed., Springer-Verlag, Berlin.
Pilkey, W.D., 1994, Formulas for Stress, Strain and Structural Matrices, Wiley, New York.
Wunderlich, W., 1976., Zur computerorientierten Formulierung von Stabilit atsproblemen, (Computer
oriented Forumulation of Problems in Stability), in Festschrift W. Zerna, Institut KIB, Werner-
Verlag, D usseldorf, pp. 111-119.
Wunderlich, W. and Beverungen, G., 1977, Geometrisch nichtlineare Theorie und Berechnung eben
gekr ummter Stabe (Geometrically nonlinear theory and analysis of plane, curved rods),
Bauingenieur, Vol. 52, pp. 225-237.
Ziegler, H., 1977, Principles of Structural Stability, 2nd ed., Birkh auser Verlag Basel and Stuttgart.
Zurm uhl, R., 1963, Praktische Mathematik, 4th ed., Springer-Verlag, Berlin.
Problems
dx 4
dx 2
11.1 Start with the governing differential equation EId 4
Nd 2
w/
+
w/
=
p z and
derive the stiffness matrix for a beam subject to an axial load N .
Hint:
First derive a transfer matrix [Eq. (11.47)] and then convert it to a stiffness
matrix.
Answer: Eq. (11.50)
11.2 Consider a column free on one end and fixed on the other. Distinguish between first
and second order theory analyses.
11.3 Find the axial force P cr that will buckle a column that is fixed at one end and hinged
at the other.
L 2
Answer:
P cr =
20
.
19 EI
/
11.4 A column is fixed at one end and guided at the other end. Find the critical axial force.
Indicate how the critical load changes if a hinged support replaces the fixed end.
2 EI
L 2 , Hinged end P cr = π
2 EI
4 L 2
Answer:
Fixed end P cr = π
/
/(
)
11.5 A horizontal beam of length 2 L is hinged at both ends and rests on a rigid support
at x
=
1
.
5 L from the left end. Find the critical axial force.
L 2
Answer:
P cr
=
5
.
89 EI
/
11.6 Suppose the moment of inertia of a beam of length L varies with the axial coordinate x
as I
is a known constant and I 0 is a nominal moment
of inertia. Use Galerkin's method to find the critical axial load. Begin with the trial
solution
=
I 0
(
1
+ β
sin
π
x
/
L
)
, where
β
m
π
sin i
x
L
w =
i
=
1
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