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u1 u2 1 2
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In matrix form , Where , [p]= parametric matrix II. Displacement function in terms of nodal displacement Express displacement function in terms of nodal displacement using the co-ordinates of nodes Where ,[A]= connectivity matrix Obtained from eq. 2 and put into eq. 1 we get Where [N]= shape function [N]= III. shape function :- Inverse is obtained by using method of adjoin Sum of shape functions is always unity At node 2, x=
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Total Dof are 04 w= translation
In matrix form ------1 Where [p]= parametric matrix II. Displacement function in terms of nodal displacement Express displacement function in terms of nodal displacement using the coordinates of nodes x=0 at node 1 and x=L at node 2 Where [A]= connectivity matrix Obtaining from eq. 2 and put into eq. 1 We get, Where [N]= shape function III. Shape functions Inverse of [A] is obtained using elementary operations of matrix algebra At node 1;x=0 N1=1 , N2=0 , N3=0 , N4=0 At node,x=L N1=0 , N2=0 , N3=1 , N4=0
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,
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+ (x)=0 at all nodes and
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Shape functions for the four noded rectangular element in natural co-ordinates system are- Cartesian coordinates of point ‘p’ are
Given p(x,y) =p(7,4) From eq. 1 Put into the eq. 2 Put this value in eq. 3
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K=
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Suppose we want to find Now, Where, are polynomial of degree n. Note:- shape function has a unit value at one nodal point (for which it is calculated) & zero value at all other nodal points. Simple exercise-
Fig.
Variation of shape function
Fig.
2. Three nodal truss element.
Fig.
Similarly
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Equations of equilibrium State of stress at a point State of strain at a point Strain -displacement relationship Stress strain relation Que – show that the following state of stress Ps in equilibrium Sol- differential equation of equilibrium of 2D elasticity problem is given by Putting these differential quantities in equations of equilibrium 1 and satisfying those. This proves that given state of stress is in equilibrium.
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