6.65 through 6.68 An extruded beam has the cross section shown. Determine (a) the location of the shear center O, (b) the distribution of the shearing stresses caused by the vertical shearing force V shown applied at O.
Fig. p6.67
(a)
Find the location of the shear center O.
Answer to Problem 68P
The location of the shear center O is
Explanation of Solution
Calculation:
Calculate the moment of inertia as shown below.
Here, b is the width of the section, d is the height of the section, A is the area of the beam, and
Calculate the moment of inertia for whole section as shown below.
Calculate the forces acting along the member as shown below.
Here,
Sketch the cross section of flange as shown in Figure 1.
Refer to Figure 1.
Calculate the first moment of area as shown below.
Calculate the first moment of area for AB as shown below.
Calculate the horizontal shear per unit length as shown below.
Here, V is the vertical shear.
Substitute
Calculate the force
Substitute
For flange AB and flange HJ:
Substitute
For flange DE and flange FG:
Substitute
Sketch the shear flow as shown in Figure 2.
Refer to Figure 2.
Calculate the eccentricity as shown below.
Substitute
Therefore, the location of the shear center O is
(b)
Find the distribution of the shearing stresses caused by the vertical shearing force.
Answer to Problem 68P
The shearing stress at point B, E, G, and J is
The shearing stress at point A and H is
The shearing stress at point just above D and just below F is
The shearing stress at point just to the right of D and just to the right of F is
The shearing stress at point just below D and just above F is
The shearing stress at point K is
Explanation of Solution
Given information:
The vertical shear is
Calculation:
Refer to part (a).
The moment of inertia
Calculate the shear stress as shown below.
At point B, E, G, and J:
Calculate the first moment of area as shown below.
Hence, the shearing stress at point B, E, G, and J is
At point A and H:
Calculate the first moment of area as shown below.
The thickness of the section is
Calculate the shear stress as shown below.
Substitute
Hence, the shearing stress at point A and H is
At point just above D and just below F:
Calculate the first moment of area as shown below.
The thickness of the section is
Calculate the shear stress as shown below.
Substitute
Hence, the shearing stress at point just above D and just below F is
At point just to the right of D and just to the right of F:
Calculate the first moment of area as shown below.
The thickness of the section is
Calculate the shear stress as shown below.
Substitute
Hence, the shearing stress at point just to the right of D and just to the right of F is
At point just below D and just above F:
Calculate the first moment of area as shown below.
The thickness of the section is
Calculate the shear stress as shown below.
Substitute
Hence, the shearing stress at point just below D and just above F is
At point just below D and just above F:
Calculate the first moment of area as shown below.
The thickness of the section is
Calculate the shear stress as shown below.
Substitute
Hence, the shearing stress at point just below D and just above F is
At point K:
Calculate the first moment of area as shown below.
The thickness of the section is
Calculate the shear stress as shown below.
Substitute
Therefore, the shearing stress at point K is
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Chapter 6 Solutions
Mechanics of Materials, 7th Edition
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