Second Moment of Area and Maximum Stress
1
eng1201 tutorial 10 problem set (Set No. 6)
1. Calculate the area A, the location of the neutral axis, and the second moment of area (cross section stiffness) IXX for each of the following shapes, and rank them in order of increasing stiffness. Scale the dimensions from the drawings, and work in mm. For a circle,
π r4 I= . 4
X X X (b) (a) X (d)
X X X
X
(c)
X X X X X
X
(e)
(f)
(g)
X
X X
X X
X
(h) (i)
(j)
2
2. A rectangular beam with a cross section 200 mm x 100 mm spans 6 metres and carries a uniform load of 2 kN/m. a) Calculate the reactions. b) Draw the shear force and bending moment diagrams. c) Calculate I for the cross section, about the axis of bending. d) Calculate the maximum compressive stress in the beam, and show where it occurs along the length, and on the cross section. e) Calculate the maximum tensile stress in the beam, and show where it occurs along the length, and on the cross section.
2 kN/m
6 metres
3. The machine component shown below can be analysed as an overhung simply supported beam. It is made of cast iron, which is much stronger in compression than in tension. If the maximum stress must be limited to 30 MPa in tension and 100 MPa in compression, calculate the maximum value of the load P that can be applied.
200 50 200
P
50 cross section dimensions in mm 0.5 m
1.0 m
3
4. Calculate the maximum stress in a beam which has the cross section drawn below. It is subjected to a moment about the YY axis of 20 kN-m (top of the beam in compression). Is the maximum stress tensile or compressive?
50
200
50
Y 200 dimensions in mm
Y 40
5. Two 50 mm by 150 mm wooden planks are glued together to form a T section as shown in the figure below. If a bending moment of 5 kNm is applied to the beam acting around a horizontal axis (top in compression), a) Find the stresses at the extreme fibres. b) Calculate the total compressive force developed by the
eng1201 tutorial 10 problem set (Set No. 6)
1. Calculate the area A, the location of the neutral axis, and the second moment of area (cross section stiffness) IXX for each of the following shapes, and rank them in order of increasing stiffness. Scale the dimensions from the drawings, and work in mm. For a circle,
π r4 I= . 4
X X X (b) (a) X (d)
X X X
X
(c)
X X X X X
X
(e)
(f)
(g)
X
X X
X X
X
(h) (i)
(j)
2
2. A rectangular beam with a cross section 200 mm x 100 mm spans 6 metres and carries a uniform load of 2 kN/m. a) Calculate the reactions. b) Draw the shear force and bending moment diagrams. c) Calculate I for the cross section, about the axis of bending. d) Calculate the maximum compressive stress in the beam, and show where it occurs along the length, and on the cross section. e) Calculate the maximum tensile stress in the beam, and show where it occurs along the length, and on the cross section.
2 kN/m
6 metres
3. The machine component shown below can be analysed as an overhung simply supported beam. It is made of cast iron, which is much stronger in compression than in tension. If the maximum stress must be limited to 30 MPa in tension and 100 MPa in compression, calculate the maximum value of the load P that can be applied.
200 50 200
P
50 cross section dimensions in mm 0.5 m
1.0 m
3
4. Calculate the maximum stress in a beam which has the cross section drawn below. It is subjected to a moment about the YY axis of 20 kN-m (top of the beam in compression). Is the maximum stress tensile or compressive?
50
200
50
Y 200 dimensions in mm
Y 40
5. Two 50 mm by 150 mm wooden planks are glued together to form a T section as shown in the figure below. If a bending moment of 5 kNm is applied to the beam acting around a horizontal axis (top in compression), a) Find the stresses at the extreme fibres. b) Calculate the total compressive force developed by the