Question 1

For the spring shown in Figure Q1, determine the displacement at nodes 2,3 and 4 and the forces in each element given that Node 5 displaces by a known distance d.

Using the method as outlined in the lectures.

(i) Formulate the stiffness matrices for each element

(ii) Apply the boundary conditions and the compatibility condition for the deflections

(iii) Derive the equations for the balance between the external and internal (elemental) forces at each node.

(iv) Form the global stiffness matrix.

(v) Reduce the system of equations by applying the relevant boundary conditions.

(vi) Solve the system of equations for the unknown deflections and unknown force, F5, for the set of values assigned to you in the table below (you may use a spreadsheet if you wish).

(vii) Determine the force in each element

Figure Q1

# k1 N/mm k2 N/mm k3 N/mm k4 N/mm P N ? mm

1 500 500 1000 500 100 20

2 400 600 1000 600 200 15

3 600 400 900 400 150 20

4 700 500 1000 500 300 25

5 600 600 1000 600 200 30

6 800 400 1000 400 100 15

7 400 400 600 400 150 20

8 500 600 650 600 300 20

9 700 700 1000 700 200 25

10 600 600 1100 600 100 15

11 500 700 1000 500 200 10

12 400 800 1100 300 100 15

13 500 600 1000 600 150 20

14 600 700 900 700 200 25

15 700 500 800 700 150 30

Question 2

For the bar assembly shown in Figure Q2, determine the displacement at nodes 2 and 3, the reactions at the supports and the forces in each element.

Figure Q2

Using the method as outlined in the lectures.

(i) Formulate the stiffness matrices for each element

(ii) Apply the boundary conditions and the compatibility condition for the deflections

(iii) Derive the equations for the balance between the external and internal (elemental) forces at each node.

(iv) Form the global stiffness matrix.

(v) Reduce the system of equations by applying the relevant boundary conditions.

(vi) Solve the system of equations for the unknown deflections and unknown force, F5, for the set of values assigned to you in the table below (you may use a spreadsheet if you wish).

(vii) Determine the force in each element

For all elements, take A=6 x 10-4 m2 and E=200 GN/m2

# L1 (mm) L2 (mm) L3 (mm) F2 (kN) F3 (kN)

1 500 500 1000 5 1

2 400 600 1000 6 2

3 600 400 900 4 1.5

4 700 500 1000 5 3

5 600 600 1000 6 2

6 800 400 1000 4 1

7 400 400 600 4 1

8 500 600 650 6 3

9 1000 700 400 2 6

10 1000 600 500 1.5 4

11 1000 800 600 3 5

12 600 400 400 2 6

13 650 500 700 1 4

14 600 700 500 4 4

15 650 600 600 2 3

Question 3

For the truss shown in Figure Q3, determine

(i) The global stiffness matrix

(ii) the x and y displacements at Node 1

(iii) the reactions at the supports.

(iv) the forces in each element.

For all elements, E = 2x1011 Pa and A = 6x10-4m2

Figure Q3

# L1 L2 L3 H V

1 1 1 1 20 40

2 1 1.5 1 25 35

3 2 2.5 2 30 30

4 2 3.0 2 35 35

5 2 3.5 2 40 40

6 3 5 4 35 50

7 4 5 3 30 55

8 3 4 3 25 50

9 2 2 2 30 45

10 4 6 4 35 40

11 4 5 3 40 45

12 2 3 2 45 50

13 3 5 3 50 60

14 4 6 4 55 65

15 4 5 3 60 70

Question 4

For the truss shown in Figure Q4, determine

(i) The global stiffness matrix

(ii) the x displacementat Node 2

(iii) the reactions at the supports.

(iv) the forces in each element.

For all elements, E = 2x1011 Pa and A = 6x10-4m2

Figure Q4

# H kN ??mm

1 50 20

2 45 25

3 40 30

4 35 35

5 30 40

6 25 45

7 20 50

8 25 55

9 30 20

10 40 25

11 45 30

12 50 35

13 55 40

14 60 45

15 55 50

Question 5

For the truss shown in Figure Q5, determine

(i) The global stiffness matrix

(ii) the y displacement at Node 2

(iii) the reactions at the supports.

(iv) the forces in each element.

For all elements, E = 2x1011 Pa

For elements 1,2,4, and 5 A = 0.0065 m2

For element 3, A = 0.0130 m2

Figure Q5

# L1 m L2 m P kN

1 2.4 1.8 50

2 2.8 2.1 45

3 3.2 2.4 40

4 3.6 2.7 45

5 4.0 3.0 50

6 2.0 1.5 55

7 2.4 1.8 60

8 2.8 2.1 55

9 3.2 2.4 25

10 3.6 2.7 30

11 4.0 3.0 45

12 2.0 1.5 50

13 5.0 3.75 60

14 2.8 2.1 55

15 3.2 2.4 50

For the spring shown in Figure Q1, determine the displacement at nodes 2,3 and 4 and the forces in each element given that Node 5 displaces by a known distance d.

Using the method as outlined in the lectures.

(i) Formulate the stiffness matrices for each element

(ii) Apply the boundary conditions and the compatibility condition for the deflections

(iii) Derive the equations for the balance between the external and internal (elemental) forces at each node.

(iv) Form the global stiffness matrix.

(v) Reduce the system of equations by applying the relevant boundary conditions.

(vi) Solve the system of equations for the unknown deflections and unknown force, F5, for the set of values assigned to you in the table below (you may use a spreadsheet if you wish).

(vii) Determine the force in each element

Figure Q1

# k1 N/mm k2 N/mm k3 N/mm k4 N/mm P N ? mm

1 500 500 1000 500 100 20

2 400 600 1000 600 200 15

3 600 400 900 400 150 20

4 700 500 1000 500 300 25

5 600 600 1000 600 200 30

6 800 400 1000 400 100 15

7 400 400 600 400 150 20

8 500 600 650 600 300 20

9 700 700 1000 700 200 25

10 600 600 1100 600 100 15

11 500 700 1000 500 200 10

12 400 800 1100 300 100 15

13 500 600 1000 600 150 20

14 600 700 900 700 200 25

15 700 500 800 700 150 30

Question 2

For the bar assembly shown in Figure Q2, determine the displacement at nodes 2 and 3, the reactions at the supports and the forces in each element.

Figure Q2

Using the method as outlined in the lectures.

(i) Formulate the stiffness matrices for each element

(ii) Apply the boundary conditions and the compatibility condition for the deflections

(iii) Derive the equations for the balance between the external and internal (elemental) forces at each node.

(iv) Form the global stiffness matrix.

(v) Reduce the system of equations by applying the relevant boundary conditions.

(vi) Solve the system of equations for the unknown deflections and unknown force, F5, for the set of values assigned to you in the table below (you may use a spreadsheet if you wish).

(vii) Determine the force in each element

For all elements, take A=6 x 10-4 m2 and E=200 GN/m2

# L1 (mm) L2 (mm) L3 (mm) F2 (kN) F3 (kN)

1 500 500 1000 5 1

2 400 600 1000 6 2

3 600 400 900 4 1.5

4 700 500 1000 5 3

5 600 600 1000 6 2

6 800 400 1000 4 1

7 400 400 600 4 1

8 500 600 650 6 3

9 1000 700 400 2 6

10 1000 600 500 1.5 4

11 1000 800 600 3 5

12 600 400 400 2 6

13 650 500 700 1 4

14 600 700 500 4 4

15 650 600 600 2 3

Question 3

For the truss shown in Figure Q3, determine

(i) The global stiffness matrix

(ii) the x and y displacements at Node 1

(iii) the reactions at the supports.

(iv) the forces in each element.

For all elements, E = 2x1011 Pa and A = 6x10-4m2

Figure Q3

# L1 L2 L3 H V

1 1 1 1 20 40

2 1 1.5 1 25 35

3 2 2.5 2 30 30

4 2 3.0 2 35 35

5 2 3.5 2 40 40

6 3 5 4 35 50

7 4 5 3 30 55

8 3 4 3 25 50

9 2 2 2 30 45

10 4 6 4 35 40

11 4 5 3 40 45

12 2 3 2 45 50

13 3 5 3 50 60

14 4 6 4 55 65

15 4 5 3 60 70

Question 4

For the truss shown in Figure Q4, determine

(i) The global stiffness matrix

(ii) the x displacementat Node 2

(iii) the reactions at the supports.

(iv) the forces in each element.

For all elements, E = 2x1011 Pa and A = 6x10-4m2

Figure Q4

# H kN ??mm

1 50 20

2 45 25

3 40 30

4 35 35

5 30 40

6 25 45

7 20 50

8 25 55

9 30 20

10 40 25

11 45 30

12 50 35

13 55 40

14 60 45

15 55 50

Question 5

For the truss shown in Figure Q5, determine

(i) The global stiffness matrix

(ii) the y displacement at Node 2

(iii) the reactions at the supports.

(iv) the forces in each element.

For all elements, E = 2x1011 Pa

For elements 1,2,4, and 5 A = 0.0065 m2

For element 3, A = 0.0130 m2

Figure Q5

# L1 m L2 m P kN

1 2.4 1.8 50

2 2.8 2.1 45

3 3.2 2.4 40

4 3.6 2.7 45

5 4.0 3.0 50

6 2.0 1.5 55

7 2.4 1.8 60

8 2.8 2.1 55

9 3.2 2.4 25

10 3.6 2.7 30

11 4.0 3.0 45

12 2.0 1.5 50

13 5.0 3.75 60

14 2.8 2.1 55

15 3.2 2.4 50

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