Tutorial 3: Unsymmetrical Bending & 3D_Trusses
1. A beam of length 2 m has the unequal –leg angle section shown in fig. 1. It has Ix=0.8x10-6 m4
, Iy = 0.4x10-6 m4 and the angle between x-axis and the principal second moment of area axis,
u is 30°. The beam is subjected to a constant bending moment (Mx) of magnitude 1000 N.m
about the x-axis as shown.
V Y
UU
30°
X
Mx
[1x10-6, 0.2x10-6 ; -70°54’ to u, 0.2847x10-6 m4 ; 6.62 mm, 90° to NA]
Figure 1
Determine:
1.1 the values of the principal second moment of inertia IU and IV respectively;
1.2 the inclination of the NA, or line of zero stress NA relative to the U-axis, and the
value of the second moment of area of the section about the NA, i.e INA.
1.3 the magnitude, direction and sense of the resultant maximum deflection of the
centroid.
For the beam material, E=200 GPa. For a beam subjected to a constant bending moment, M,
ML2
the maximum deflection is given by the formula: δ=
8 EI
2. A cantilever beam 1 m long consists of a 60 mm x 60 mm x 10 mm angle section with the top
face AB horizontal as shown in figure 2. It carries a load of 800 N at the free end, the line of
action of the load passing through the centroid of the section and inclined at 30°
anticlockwise to the vertical.
30° Y V
B
A
X
C U
[sA= 136.5 MPa; sB= -70.4 MPa; sC= -102.0 MPa; β=85.9° ]
18.4 mm
Figure 2
Determine the stress at the corners A, B and C at the fixed end and also the position of the neutral
−6
axis. I x = I y = 0.3488 x10 m I V = 0.1472 x10 −6 m 4 I U = 0.5506 x10 −6 m 4
4
1. A beam of length 2 m has the unequal –leg angle section shown in fig. 1. It has Ix=0.8x10-6 m4
, Iy = 0.4x10-6 m4 and the angle between x-axis and the principal second moment of area axis,
u is 30°. The beam is subjected to a constant bending moment (Mx) of magnitude 1000 N.m
about the x-axis as shown.
V Y
UU
30°
X
Mx
[1x10-6, 0.2x10-6 ; -70°54’ to u, 0.2847x10-6 m4 ; 6.62 mm, 90° to NA]
Figure 1
Determine:
1.1 the values of the principal second moment of inertia IU and IV respectively;
1.2 the inclination of the NA, or line of zero stress NA relative to the U-axis, and the
value of the second moment of area of the section about the NA, i.e INA.
1.3 the magnitude, direction and sense of the resultant maximum deflection of the
centroid.
For the beam material, E=200 GPa. For a beam subjected to a constant bending moment, M,
ML2
the maximum deflection is given by the formula: δ=
8 EI
2. A cantilever beam 1 m long consists of a 60 mm x 60 mm x 10 mm angle section with the top
face AB horizontal as shown in figure 2. It carries a load of 800 N at the free end, the line of
action of the load passing through the centroid of the section and inclined at 30°
anticlockwise to the vertical.
30° Y V
B
A
X
C U
[sA= 136.5 MPa; sB= -70.4 MPa; sC= -102.0 MPa; β=85.9° ]
18.4 mm
Figure 2
Determine the stress at the corners A, B and C at the fixed end and also the position of the neutral
−6
axis. I x = I y = 0.3488 x10 m I V = 0.1472 x10 −6 m 4 I U = 0.5506 x10 −6 m 4
4
