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One Dimensional ProblemsOne Dimensional ProblemsP. M. Agrawal

V. J. Patel

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1D Problem1D Problem

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Fig. 1

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1D Problem1D Problem

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Fig. 1

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1D Problem1D Problem

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Fig. 2

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Element ConnectivityElement Connectivity

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Fig. 3

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Natural Coordinate SystemNatural Coordinate System

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Fig. 4

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Shape FunctionShape Function

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Fig. 5

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P. E. in 1D ProblemP. E. in 1D Problem

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(21)

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Element Stiffness MatrixElement Stiffness Matrix

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Force TermForce Term

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Force TermForce TermSimilarly,

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At this stage, element stiffness matrices ke, fe, and Te have been obtained. After

we account for the element connectivity (fig. 3), the total potential energy in

Eq. 21 can be written as

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Properties of KProperties of K The dimension of the global stiffness K is

(N x N), where N is the number of nodes.This follows from the fact that each node

has only one degree of freedom.

K is symmetric.

K is a banded matrix. That is, all elements

outside of the band are zero.

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ExampleExample Consider the thin steel plate in Fig. The plate has

a uniform thickness t = 25 mm, Youngs modulus E

= 200 GPa, and weight density = 7850 kg/m3. Inaddition to its self-weight, the plate is subjectedto a point load P = 100 KN at its mid point.1. Model the late with two finite elements.

2. Write down expressions for the element stiffnessmatrices and element body force vectors.

3. Assemble the global stiffness matrix K and globalload vector F.

4. Solve for the global displacement vector Q.5. Evaluate the stresses in each element.

6. Determine the reaction force at the support.

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Since, Q1 =0

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ExerciseExerciseConsider the bar in Fig. 8 loaded as shown. Determine the nodal

displacement, element stresses, and support reactions.

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