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CEE457 Advanced Structures I, Winter 2015HW 2 Due 1/28/15, 5:00 pm

Problem 0: Read the following sections of the Logan text: 3.1, 3.2, 3.10, 3.11, 3.12, 3.13.

Problem 1: Determine the exact displacement field, ( )exact u x , for the tapered truss shown below. This requiresfinding the analytical solution to the differential equation that defines equilibrium at a point on the element:

( ) 0exact dud

A x E dx dx

use the boundary conditions:

100 0 50 0 0 0 0

Hint:

Problem 2: Use the finite element method with meshes of 1, 2 and 4 two-node constant-area truss elements to

determine the displacement and stress fields for the axially loaded bar shown in Problem 1. Provide sketches ofyour models with nodal displacements and reaction forces identified. Your constant-area truss elements shouldhave an area equal to the true area of the bar at the center of the element. For example, for the two-elementmesh, the area of the elements should be 1.1875 in 2 and 2.6875 in 2.

Problem 3: Find the approximate stiffness equations f kd for the tapered truss element shown below thathas an elastic modulus, E. The area of this truss is defined 1 . To derive the stiffnessequations, use the Principal of Minimum Potential Energy and assume a linear displacement field within the

element: 1 2 1

x xu x N N d d

L L

. Your final set of equations should be in terms of f , d , E, L,

A1 and A2.

1 1 1 4 where 0,100 and L = 100 in.

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Problem 4: Redo Problem #2 using the element stiffness equations that you developed in Problem #3 andmeshes of 1 and 2 tapered-area two-node truss elements.

Problem 5: Create two plots that show 1) the displacement field and 2) the stress field along the length of thetapered bar from Problems 1-4. Each plot should show i) the exact solution from Problem #1, ii) the solution asdetermined using meshes of 1, 2 and 4 constant-area two-node truss elements from Problem #2 and iii) thesolution as determined using meshes of 1 and 2 tapered-area two-node truss elements from Problem #4.Remember that for the two-node truss elements, the displacement field is determined by the shape functions andthe nodal displacements and the stress field is determined by the derivatives of the shape functions, the elasticmodulus, and the nodal displacements.

Problem 6: Problem 3.14 from Logan:

Consider the following displacement function for the two-noded bar element:

u(x) = a + bx 2

Is this a valid displacement function? Discuss why or why not.

Problem 7: Consider the following shape functions, N , for use in deriving the stiffness equations (f = kd) for atwo-node truss element that spans x=0 to x=L. The approximate displacement function for the element would bedefined as d N x u )( where d is the 2x1 vector of nodal displacements.

L x

L x

x N x N N 141

143

)()( 21

Do these shape functions satisfy the requirements described in class (continuity between elements, continuitywithin the element, completeness, and ability to represent boundary conditions)? For each of these requirements,describe the requirement and then explain why the shape functions satisfy or do not satisfy the requirement

Problem 8: a) Determine the exact displacement field and stress field for the 1D problem shown below. b) Determine the displacement and stress fields using the finite element method with meshes that comprise 1 2-node element and 2 2-node elements. c) Determine displacement and stress fields using the finite elementmethod with a mesh comprising 1 3-node element. d) Plot the displacement fields and stress fields from parts a, band c. You do not need to re-derive the stiffness matrix and equations for the truss elements; however, you willneed to compute equivalent nodal loads.

Hint: 1+ constant

x

1 1 , x xd f 2 2

, x xd f

0 x x L

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