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A. Channel section
Cantilever beam is loaded with lateral force aligned with the y-axis (see figure below). Use general XY axis method to define bending stresses and displacement. Note: Use g=10 m/s2 for your calculations. The beam is made from 6082-T6 Aluminium alloy E = 70,000N/mm2, ν = 0.3.
1. Calculate θp, the angle between the beam axis x and principal axis 1. Give your answer in rad.
2. Define principal second moment of area I11 relative to the principal axis 1 passing through the
section centroid. Give your answer in mm4.
3. Define principal second moment of area I22 relative to the principal axis 2 passing through the
section centroid. Give your answer in mm4.
4. Define principal product second moment of area I12 in the coordinate system 1-2 passing through
the section centroid. Give your answer in mm4.
5. Calculate the internal moment Mx at z=0 mm. Give your answer in N mm.
6. Calculate the internal moment My at z=0 mm. Give your answer in N mm.
7. Calculate the bending stress σz at point O. Give your answer in N/mm2.
8. Calculate the bending stress σz at point A. Give your answer in N/mm2.
9. Calculate the deflection of the beam. Give your answer in mm.
B. Rectangular section at 45° to loading
Cantilever beam is loaded with lateral force aligned with the y’-axis (see figure below). Use Principal 1-2 axis method to define bending stresses and displacement. Note: Use g=10 m/s2 for your calculations. You can choose any load from your Structures Lab of the corresponding experiment, but give your answers per unit load (i.e. answer / chosen load). The beam is made from 6082-T6 Aluminium alloy E = 70,000N/mm2, ν = 0.3.
1. Calculate θp, the angle between the beam axis x’ and principal axis 1. Give your answer in rad.
2. Calculate the internal moment Mx’ at z=0 mm. Normalise the obtained value with respect to
the applied force. Give your answer in N mm/N.
3. Calculate the internal moment My’ at z=0 mm. Normalise the obtained value with respect to
the applied force. Give your answer in N mm/N.
4. Calculate the moment M1 with respect to principal axis 1-2. Normalise the obtained value with
respect to the applied force. Give your answer in N mm/N.
5. Calculate the moment M2 with respect to principal axis 1-2. Normalise the obtained value with
respect to the applied force. Give your answer in N mm/N.
6. Calculate the bending stress σz at point O. Normalise the obtained value with respect to the
applied force. Give your answer in N/mm2 /N.
7. Calculate the bending stress σz at point A. Normalise the obtained value with respect to the
applied force. Give your answer in N/mm2 /N.
8. Calculate the bending stress σz at point B. Normalise the obtained value with respect to the
applied force. Give your answer in N/mm2 /N.
9. Calculate the bending stress σz at point D. Normalise the obtained value with respect to the
applied force. Give your answer in N/mm2 /N.
10. Calculate δ1, the component of the deflection in the principal coordinate system of the beam.
Normalise the obtained value with respect to the applied force. Give your answer in mm/N.
11. Calculate δ2, the component of the deflection in the principal coordinate system of the beam.
Normalise the obtained value with respect to the applied force. Give your answer in mm/N.
12. Calculate the magnitude of the deflection vector δ. Normalise the obtained value with respect
to the applied force. Give your answer in mm/N.
13. Calculate the direction of the deflection δ (resultant of δ1 and δ2) defined by the angle φ. The
angle φ is defined in the counter-clockwise direction from the principal axis 1. Give your answer in rad.
14. Calculate δx’ (the projection of the deflection vector δ on the x’ axis). Normalise the obtained value with respect to the applied force. Give your answer in mm/N.
15. Calculate δy’ (the projection of the deflection vector δ on the y’ axis). Normalise the obtained value with respect to the applied force. Give your answer in mm/N.
C. L-section
Cantilever beam is loaded with lateral force aligned with the y-axis (see figure below). Use Neutral axis (NA) method to define bending stresses and displacement. Note: Use g=10 m/s2 for your calculations. The beam is made from 6082-T6 Aluminium alloy E = 70,000N/mm2, ν = 0.3.
1. Calculate θp, the angle between the beam axis x and principal axis 1. Give your answer in rad.
2. Define principal second moment of area I11 relative to the principal axis 1 passing through the
section centroid. Give your answer in mm4.
3. Define principal second moment of area I22 relative to the principal axis 2 passing through the
section centroid. Give your answer in mm4.
4. Define principal product second moment of area I12 in the coordinate system 1-2 passing
through the section centroid. Give your answer in mm4.
5. Calculate the internal moment Mx at z=0 mm. Give your answer in N mm.
6. Calculate the internal moment My at z=0 mm. Give your answer in N mm.
7. Calculate the angle α between NA and axis x. Give your answer in rad.
8. CalculateMNAthecomponentontotheNAaxisoftheinternalmoment.Giveyouranswerin
N mm.
9. Calculate the bending stress σz at point A. Give your answer in N/mm2.
10. Calculate the bending stress σz at point B. Give your answer in N/mm2.
11. Calculate the bending stress σz at point D. Give your answer in N/mm2.
12. Calculate δ1, the component of the deflection in the principal coordinate system of the beam.
Give your answer in mm.
13. Calculate δ2, the component of the deflection in the principal coordinate system of the beam.
Give your answer in mm.
14. Calculate the magnitude of the deflection vector δ. Give your answer in mm.
15. Calculate the direction of the deflection δ (resultant of δ1 and δ2) defined by the angle φ. The
angle φ is defined in the counter-clockwise direction from the principal axis 1. Give your
answer in rad.
16. Calculate δx (the projection of the deflection on x axis). Give your answer in mm.
17. Calculate δy (the projection of the deflection on y axis). Give your answer in mm.
D. Lipped L-section
Simply supported beam is loaded with lateral force aligned with the y-axis (see figure below). Use
Principal 1-2 axis method to define bending stresses and displacement. Note: Use g=10 m/s2 for your calculations. You can choose any load from your Structures Lab of the corresponding experiment, but give your answers per unit load (i.e. answer / chosen load). The beam is made from 6082-T6 Aluminium alloy E = 70,000N/mm2, ν = 0.3.
1. Calculate θp, the angle between the beam axis x and principal axis 1. Give your answer in rad.
2. Define principal second moment of area I11 relative to the principal axis 1 passing through the
section centroid. Give your answer in mm4.
3. Define principal second moment of area I22 relative to the principal axis 2 passing through the
section centroid. Give your answer in mm4.
4. Define principal product second moment of area I12 in the coordinate system 1-2 passing
through the section centroid. Give your answer in mm4.
5. Calculate the internal moment Mx at z=520 mm. Normalise the obtained value with respect to
the applied force. Give your answer in N mm/N.
6. Calculate the internal moment My at z=520 mm. Normalise the obtained value with respect to
the applied force. Give your answer in N mm/N.
7. Calculate the moment M1 with respect to principal axis 1-2. Normalise the obtained value with
respect to the applied force. Give your answer in N mm/N.
8. Calculate the moment M2 with respect to principal axis 1-2. Normalise the obtained value with
respect to the applied force. Give your answer in N mm/N.
9. Calculate the bending stress σz at point O. Normalise the obtained value with respect to the
applied force. Give your answer in N/mm2/N.
10. Calculate the bending stress σz at point A. Normalise the obtained value with respect to the
applied force. Give your answer in N/mm2/N.
11. Calculate the bending stress σz at point B. Normalise the obtained value with respect to the
applied force. Give your answer in N/mm2/N.
12. Calculate the bending stress σz at point D. Normalise the obtained value with respect to the
applied force. Give your answer in N/mm2/N.
13. Calculate δ1, the component of the deflection in the principal coordinate system of the beam.
Normalise the obtained value with respect to the applied force. Give your answer in mm/N.
14. Calculate δ2, the component of the deflection in the principal coordinate system of the beam.
Normalise the obtained value with respect to the applied force. Give your answer in mm/N.
15. Calculate the magnitude of the deflection vector δ. Normalise the obtained value with respect
to the applied force. Give your answer in mm/N.
16. Calculate the direction of the deflection δ (resultant of δ1 and δ2) defined by the angle φ. The angle φ is defined in the counter-clockwise direction from the principal axis 1. Give your answer in rad.
17. Calculate δx (the projection of the deflection on x axis). Normalise the obtained value with respect to the applied force. Give your answer in mm/N.
18. Calculate δy (the projection of the deflection on y axis). Normalise the obtained value with respect to the applied force. Give your answer in mm/N.

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