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(ii) Cubic shape functions
N _ {1} ^ {(e)} = \frac {\left(x ^ {(e)} - x _ {2} ^ {(e)}\right) \left(x ^ {(e)} - x _ {3} ^ {(e)}\right) \left(x ^ {(e)} - x _ {4} ^ {(e)}\right)}{\left(x _ {1} ^ {(e)} - x _ {2} ^ {(e)}\right) \left(x _ {1} ^ {(e)} - x _ {3} ^ {(e)}\right) \left(x _ {1} ^ {(e)} - x _ {4} ^ {(e)}\right)}
N _ {2} ^ {(e)} = \frac {\left(x ^ {(e)} - x _ {1} ^ {(e)}\right) \left(x ^ {(e)} - x _ {3} ^ {(e)}\right) \left(x ^ {(e)} - x _ {4} ^ {(e)}\right)}{\left(x _ {2} ^ {(e)} - x _ {1} ^ {(e)}\right) \left(x _ {2} ^ {(e)} - x _ {3} ^ {(e)}\right) \left(x _ {2} ^ {(e)} - x _ {4} ^ {(e)}\right)}
N _ {3} ^ {(e)} = \frac {\left(x ^ {(e)} - x _ {1} ^ {(e)}\right) \left(x ^ {(e)} - x _ {2} ^ {(e)}\right) \left(x ^ {(e)} - x _ {4} ^ {(e)}\right)}{\left(x _ {3} ^ {(e)} - x _ {1} ^ {(e)}\right) \left(x _ {3} ^ {(e)} - x _ {2} ^ {(e)}\right) \left(x _ {3} ^ {(e)} - x _ {4} ^ {(e)}\right)}
N _ {4} ^ {(e)} = \frac {\left(x ^ {(e)} - x _ {1} ^ {(e)}\right) \left(x ^ {(e)} - x _ {2} ^ {(e)}\right) \left(x ^ {(e)} - x _ {3} ^ {(e)}\right)}{\left(x _ {4} ^ {(e)} - x _ {1} ^ {(e)}\right) \left(x _ {1} ^ {(e)} - x _ {2} ^ {(e)}\right) \left(x _ {4} ^ {(e)} - x _ {3} ^ {(e)}\right)} \tag {5.59}
For the quadratic and cubic elements use 2-point and 3-point Gauss-Legendre integration rules respectively.
5.2 Develop a layered finite element Timoshenko beam program which allows for combined in-plane and bending behaviour of axially loaded beams or beams with cross-sections which are nonsymmetric about the neutral axis. Choose a displacement representation of the form
\bar {u} (x, z) = u _ {0} (x) - z \theta_ {x} (x) \tag {5.60}
in which u_{0}(x) is the axial displacement at the neutral axis.
5.3 Use the concepts developed in Chapters 4 and 5 to develop the necessary relationships for layered and nonlayered elastoviscoplastic Timoshenko beam analysis.
5.4 (i) Evaluate the additional stiffness terms required to represent the Winkler foundation by a 2-node linear Timoshenko beam element. For a foundation modulus k note that the additional virtual work term associated with the elastic foundation is
\int_ {0} ^ {l} \delta w k w d x
in which \delta w is the virtual lateral displacement.
(ii) Modify programs TIMOSH and TIMLAY to allow for beams on elastic foundations.
(iii) Use the program to analyse a uniformly loaded, simply supported beam on a Winkler foundation. The elastic closed form solution for an Euler–Bernoulli beam predicts lateral displacements
w = \sum_ {n = 1, 3, 5, \dots} ^ {\infty} \frac {4 q L ^ {4} / \left(n ^ {5} \pi^ {5} E I\right)}{1 + k L ^ {4} / \left(n ^ {4} \pi^ {4} E I\right)} \sin \frac {n \pi x}{L} \tag {5.61}
and bending moments
M = \sum_ {n = 1, 3, 5, \dots} ^ {\infty} \frac {4 q L ^ {2} / (n \pi) ^ {3}}{1 + k L ^ {4} / (n ^ {4} \pi^ {4} E I)} \sin \frac {n \pi x}{L}. \tag {5.62}
Compare the elastic results from the modified programs with the above solution for various values of kL^{4}/EI and t/L where EI is the flexural rigidity, t is the thickness and L is the length of the beam.
(iv) For a given yield stress, \sigma_{0} , evaluate the ultimate load for various values of kL^{4}/EI and t/L.
5.5 (i) Consider the problem of finding the elastic deflections of a simply supported beam of length L, flexural rigidity EI, shear rigidity GA which is subjected to a uniform load q. The beam is elastically supported at mid-span by a single linear spring of stiffness K. Modify programs TIMOSH and TIMLAY to solve this problem. Check your finite element solutions by noting that the elastic Euler–Bernoulli solution is given as
\begin{array}{l} w = \frac {4 q L ^ {4}}{E I} \sum_ {n = 1, 3, 5, \dots} ^ {\infty} \frac {\sin (n \pi x / L)}{n ^ {5}} \\ - \frac {2 K S L ^ {3}}{\pi^ {4} E I} \sum_ {n = 1, 3, 5, \dots} ^ {\infty} \left(\frac {\sin (n \pi / 2) \sin (n \pi x / L)}{n ^ {4}}\right) \tag {5.63} \\ \end{array}
in which
S = \frac {5 q L ^ {4}}{3 8 4 E I} / \left(1 + \frac {K L ^ {3}}{4 8 E I}\right). \tag {5.64}
(ii) When the load carried by the elastic support reaches a value F the supported beam becomes perfectly plastic. How can this be catered for in the modified version of TIMOSH and TIMLAY?
5.6 Use program TIMLAY to examine the effects of choosing
(i) different load incrementations
(ii) various convergence tolerances
(iii) various numbers of layers
on the example given in Section 5.4 and also Problems 5.4 and 5.5.
5.7 References
- HINTON, E. and OWEN, D. R. J., An Introduction to Finite Element Computations, Pineridge Press, Swansea, U.K., 1979.
- HUGHES, T. J. R., TAYLOR, R. L. and KANOKNUKULCHAI, S., A simple and efficient finite element for bending, Int. J. Num. Meth. Engng., 11, 1529–1543 (1977).
- COWPER, G. R., The shear coefficient in Timoshenko's Beam Theory, J. Appl. Mech., 33, 335 (1966).
- DYM, C. L. and SHAMES, I. H., Solid Mechanics: A Variational Approach, McGraw-Hill, New York, 1973.
- HINTON, E. and OWEN, D. R. J., Finite Element Programming, Academic Press, London, 1977.
Part II
Chapter 6 Preliminary theory and standard subroutines for two-dimensional elasto-plastic applications
6.1 Introduction
In Part II of this text we extend the concepts and techniques developed in Part I for one-dimensional situations to now permit the solution of two-dimensional problems. In particular the following applications are presented:
● Chapter 7 discusses the solution of elasto-plastic problems conforming to either plane stress, plane strain or axially symmetric conditions.
- Chapter 8 deals with plane stress/strain and axisymmetric problems where the material exhibits a time-dependent elasto-viscoplastic behaviour.
● Chapter 9 covers elasto-plastic plate bending situations.
The nonlinear algorithms developed in Chapter 2 will be employed in solution. These processes are general and the main modifications necessary are those appropriate to two-dimensional continuum theory or plate bending expressions which must now be used. For example the level of initial yielding will now be dependent on three or more independent stress components in place of the uniaxial case considered earlier.
The development of an elasto-plastic stress analysis program requires all of the basic features of the corresponding elastic program. In particular the same basic element formulation is employed and a wide choice of element types is available. In this text we consider three different element types all based on an isoparametric formulation. The elements included are illustrated in Fig. 6.1 and are:
- The 4-node isoparametric quadrilateral element with linear displacement variation, Fig. 6.1(a).
- The 8-node Serendipity quadrilateral element with curved sides and a quadratic variation of the displacement field within the element, Fig. 6.1(b).
- The 9-node Lagrangian quadrilateral element which additionally has a central node, Fig. 6.1(c).
The basic theoretical expressions for these elements are provided in Section 6.3. The use of these higher order elements leads to particularly efficient
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4 3 η ξ 1 2
N _ {i} (\xi , \eta) = \frac {1}{4} (1 + \xi \xi_ {i}) (1 + \eta \eta_ {i}).
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| Local node number | $\xi_i$ | $\eta_i$ |
| 1 | -1 | -1 |
| 2 | 1 | -1 |
| 3 | 1 | 1 |
| 4 | -1 | 1 |
Fig. 6.1(a) The 4-node isoparametric quadrilateral element and shape functions.
elasto-plastic solution packages. In order to simplify matters as much as possible consideration is restricted to isotropic situations.*
For all the plasticity applications presented in this text the classical incremental theory is employed with the full elasto-plastic material response being reproduced. Thus we are not concerned with limit state behaviour as predicted by rigid-plastic theories, etc.
Consideration is limited to small deformation situations where the strains can be assumed to be infinitesimal and Lagrangian and Eulerian geometric descriptions then coincide.
- Extension to orthotropic situations is feasible and has indeed been dealt with in Ref. 1.
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7 6 5 8 η ξ 1 2 3
8-node Serendipity element
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7 6 5 η 8 9 ξ 4 1 2 3
9-node Lagrangian element
- for corner nodes
N _ {i} ^ {(e)} = \frac {1}{4} (1 + \xi \xi_ {i}) (1 + \eta \eta_ {i}) (\xi \xi_ {i} + \eta \eta_ {i} - 1), \quad i = 1, 3, 5, 7,
• for midside nodes
N _ {i} ^ {(e)} = \frac {\xi_ {i} ^ {2}}{2} (1 + \xi \xi_ {i}) (1 - \eta^ {2}) + \frac {\eta_ {i} ^ {2}}{2} (1 + \eta \eta_ {i}) (1 - \xi^ {2}), \quad i = 2, 4, 6, 8.
$$<table><tr><td>Local node number</td><td> $\xi$ </td><td> $\eta$ </td></tr><tr><td>1</td><td>-1</td><td>-1</td></tr><tr><td>2</td><td>0</td><td>-1</td></tr><tr><td>3</td><td>1</td><td>-1</td></tr><tr><td>4</td><td>1</td><td>0</td></tr><tr><td>5</td><td>1</td><td>1</td></tr><tr><td>6</td><td>0</td><td>1</td></tr><tr><td>7</td><td>-1</td><td>1</td></tr><tr><td>8</td><td>-1</td><td>0</td></tr><tr><td>9</td><td>0</td><td>0</td></tr></table>
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Fig. 6.1(b) The 8-node Serendipity quadrilateral element.
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\- for corner nodes
N ^ {(n)} = \frac {1}{4} (\xi^ {2} + \xi \xi_ {i})) (\eta^ {2} + \eta \eta_ {i}), \quad i = 1, 3, \quad 5, \quad 7,
\- for midside nodes
N _ {i} ^ {(s)} = \frac {1}{2} \eta_ {i} ^ {2} (\eta^ {2} - \eta \eta_ {i}) (1 - \xi^ {2}) + \frac {1}{2} \xi_ {i} ^ {2} (\xi^ {2} - \xi \xi_ {i}) (1 - \eta^ {2}), \quad i = 2, 4, 6, 8,
\- for central node
N _ {i} ^ {(e)} = (1 - \xi^ {2}) (1 - \eta^ {2}).
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Fig. 6.1(c) The 9-node Lagrangian quadrilateral element.