Concept explainers
8.9 through 8.14 Each of the following problems refers to a rolled-steel shape selected in a problem of Chap. 5 to support a given loading at a minimal cost while satisfying the requirement σm ≤ σall. For the selected design, determine (a) the actual value of σm in the beam, (b) the maximum value of the principal stress σmax at the junction of a flange and the web.
8.14 Loading of Prob. 5.78 and selected S460 × 81.4 shape.
Fig. P5.78
(a)
The actual value of
Answer to Problem 14P
The actual value of
Explanation of Solution
Given information:
Refer to problem 5.78 in chapter 5 in the textbook.
The allowable normal stress of the beam is
Calculation:
Design of beam:
Show the free-body diagram of the beam as in Figure 1.
Determine the vertical reaction at point D by taking moment at point A.
Determine the vertical reaction at point A by resolving the vertical component of forces.
Shear force:
Show the calculation of shear force as follows;
Show the calculated shear force values as in Table 1.
Location (x) m | Shear force (V) kN |
A | 65 |
B (Left) | 65 |
B (Right) | 5 |
C (Left) | 5 |
C (Right) | –35 |
D | –35 |
Plot the shear force diagram as in Figure 2.
Bending moment:
Show the calculation of the bending moment as follows;
Show the calculated bending moment values as in Table 2.
Location (x) m | Bending moment (M) kN-m |
A | 0 |
B | 162.5 |
C | 175 |
D | 0 |
Plot the bending moment diagram as in Figure 3.
Refer to the Figure 3;
The maximum bending moment in the beam is
Write the section a property for a
Dimension | Unit( |
d | 457 |
152 | |
17.6 | |
Here, d is depth of the section,
Find the value of C using the relation:
Substitute
Find the maximum value of normal stress
Here,
Substitute
Thus, the actual value of
(b)
The maximum value of principal stress
Answer to Problem 14P
The maximum value of principal stress
Explanation of Solution
Calculation:
Find the value
Here, c is the centroid and
Substitute
Find the area of flange
Here,
Substitute
Find the centroid of flange
Substitute
Find the first moment about neutral axis
Here,
Substitute
At section C,
Find the value of
Here, actual value of normal stress
Substitute
Find the shear stress at
Modify Equation (8).
Substitute
Find the maximum shearing stress (R) using the relation:
Here,
Substitute
Determine the maximum value of the principle stress using the relation:
Here, R is the maximum shearing stress and
Substitute
At section B,
Find the maximum value of normal stress
Here,
Substitute
Find the value of
Substitute
Find the shear stress at b
Substitute
Refer to figure 3.
Substitute
Find the maximum shearing stress (R) using the relation:
Here,
Substitute
Determine the maximum value of the principle stress using the relation:
Here, R is the maximum shearing stress and
Substitute
Based on results,
Select the maximum value of principal stress
Thus, the maximum value of principal stress
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