Concept explainers
(a) Knowing that σall = 24 ksi and σall = 14.5 ksi, select the most economical wide-flange shape that should be used to support the loading shown. (b) Determine the values to be expected for τm, τm, and the principal stress σmax at the junction of a flange and the web of the selected beam.
Fig. P8.65
(a)
Select the most economical wide flange shape section.
Answer to Problem 65RP
The most economical wide flange shape section is
Explanation of Solution
Given information:
The allowable bending stress
The shear stress
Calculation:
Sketch the free body diagram of beam as shown in figure 1.
Here,
Calculate the reaction for the given structure:
Taking moment about D,
Summation of vertical force zero.
Calculate the shear force diagram as follows:
Shear force at A is
Shear force at B,
Shear force at C,
Shear force at D,
Maximum shear force occurs at the point where
Consider a section
Calculate the bending moment as follows:
Bending moment at point A,
Bending moment at point B,
Bending moment at point C,
Bending moment at point D,
Sketch the shear force and bending moment diagram as shown in Figure 2.
Refer to Figure 2.
The maximum bending moment is
Find the value of
Here,
Substitute
Write the section properties shape as shown in Table 2.
Shape | |
38.4 | |
29.0 | |
33.4 | |
32.4 | |
31.2 |
Select the section
The most economical wide flange shape section is
Write the section properties wide flange section as shown in Table 3.
Shape | W14x22 |
Area, A | |
Depth, d | |
Web thickness, | 0.230 in. |
Width | 5.000 in. |
Thickness | 0.335 in. |
Find the area of web
Here, d is the depth of the section and
Substitute
(b)
The value to be expected for
Answer to Problem 65RP
The normal stress
The shear stress
The maximum principal stress is
Explanation of Solution
Calculation:
Point E:
Find the normal stress at point E using the relation:
Here,
Substitute
Thus, the normal stress
Find the value of C using the relation:
Here, d is the depth of section.
Substitute
Find the value of
Here,
Substitute
Find the normal stress
Here,
Substitute
Find the shear stress
The point E is located at top. Since Q is zero.
Thus, the shear stress at point
Point C:
Find the normal stress at point E using the relation:
Here,
Substitute
Find the shear stress point C using the relation:
Here, V is shear force and
Substitute
Thus, the shear stress
Find the
Substitute
Find the R using the relation:
Here,
Substitute
Find the maximum principal stress using the relation:
Substitute
Compare the results,
Select the maximum value of stress for Point B is controls.
Thus, the maximum principal stress is
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