Question

4) See the 1-D structure with 4 nodes and 3 elements numbered as shown. Elements 1, 2, and 3 are have identical area $A = .01 \text{ m}^2$, and modulus $E = 100 \text{ GPa}$, but different length ($L_1 = 0.3 \text{ m}$, $L_2 =$ $0.5 \text{ m}$, $L_3 = 0.2 \text{ m}$). Due to some frictional interaction with its surroundings, element 2 is exposed to a constant distributed load $p(x) = $ $10 \text{ kN/m}$. The right side of the structure has applied load $F = 20 \text{ kN}$. Use some sort of well-documented programming to solve this problem (Matlab, Maple, or similar). Print all coding and include in your submission. a) Form the [B] and [D] element matrices. (Don't forget the [B] matrices depend on element length) Use these to get the individual element stiffness matrices $[K^e]$. b) Assemble the 3 individual [2x2] element stiffness matrices into the global [4x4] stiffness matrix $[K]$. c) Form the global RHS forcing vector. Big hint: because the distributed load is constant here, and we are considering linear interpolation, it is easy to account for: Simply calculate the resultant force, and the then apply half of that resultant force to node 2 and half to node 3. (Easy!) d) Follow procedures in the notes to adjust your $[K] \mathbf{u} = \mathbf{f}$ system of equations for the displacement B.C. at the left end ($x=0$). e) Solve for the nodal displacements. f) Now that you have displacements, use your [B] and [D] matrices to get the stress in each element.

          4) See the 1-D structure with 4 nodes and 3 elements numbered as shown. Elements 1, 2, and 3 are
have identical area $A = .01 \text{ m}^2$, and modulus $E = 100 \text{ GPa}$, but different length ($L_1 = 0.3 \text{ m}$, $L_2 =$
$0.5 \text{ m}$, $L_3 = 0.2 \text{ m}$). Due to some
frictional interaction with its
surroundings, element 2 is exposed to a
constant distributed load $p(x) = $
$10 \text{ kN/m}$. The right side of the structure
has applied load $F = 20 \text{ kN}$.
Use some sort of well-documented programming to solve this problem (Matlab, Maple, or similar).
Print all coding and include in your submission.
a) Form the [B] and [D] element matrices. (Don't forget the [B] matrices depend on element length)
Use these to get the individual element stiffness matrices $[K^e]$.
b) Assemble the 3 individual [2x2] element stiffness matrices into the global [4x4] stiffness matrix
$[K]$.
c) Form the global RHS forcing vector. Big hint: because the distributed load is constant here, and
we are considering linear interpolation, it is easy to account for: Simply calculate the resultant force,
and the then apply half of that resultant force to node 2 and half to node 3. (Easy!)
d) Follow procedures in the notes to adjust your $[K] \mathbf{u} = \mathbf{f}$ system of equations for the displacement
B.C. at the left end ($x=0$).
e) Solve for the nodal displacements.
f) Now that you have displacements, use your [B] and [D] matrices to get the stress in each element.

4) See the 1-D structure with 4 nodes and 3 elements numbered as shown. Elements 1, 2, and 3 are
have identical area A = .01 m^2, and modulus E = 100 GPa, but different length (L1 = 0.3 m, L2 =
0.5 m, L3 = 0.2 m). Due to some
frictional interaction with its
surroundings, element 2 is exposed to a
constant distributed load p(x) =
10 kN/m. The right side of the structure
has applied load F = 20 kN.
Use some sort of well-documented programming to solve this problem (Matlab, Maple, or similar).
Print all coding and include in your submission.
a) Form the [B] and [D] element matrices. (Don't forget the [B] matrices depend on element length)
Use these to get the individual element stiffness matrices [K^e].
b) Assemble the 3 individual [2x2] element stiffness matrices into the global [4x4] stiffness matrix
[K].
c) Form the global RHS forcing vector. Big hint: because the distributed load is constant here, and
we are considering linear interpolation, it is easy to account for: Simply calculate the resultant force,
and the then apply half of that resultant force to node 2 and half to node 3. (Easy!)
d) Follow procedures in the notes to adjust your [K] 𝐮 = 𝐟 system of equations for the displacement
B.C. at the left end (x=0).
e) Solve for the nodal displacements.
f) Now that you have displacements, use your [B] and [D] matrices to get the stress in each element.

Added by Paul C.

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