10.A structural steel wide flange I-beam is to be designed to have a factor of s
ID: 2073443 • Letter: 1
Question
10.A structural steel wide flange I-beam is to be designed to have a factor of safety equal to 3.0 against yielding. Given the shear and bending moment diagrams shown below for this beam, the lightest weight beam that can support this load is: (Units in the diagrams are in feet and pounds. Neglect shear, and assume self weight has already been factored.) 6,500.00 6,300.00 Shear Diagram 900.00 ,300.00 0.00 4,100.00 4,500.00 17,200.00 Moment Diagram 2,800.00 0.00 0.00 (ft) a. W6 x 25 b. W8 x 15 C. W8 x 24 d. W10x 22 e. W16x 26Explanation / Answer
I have used the following beam section properties calculator:
http://www.amesweb.info/SectionalPropertiesTabs/StructuralSteelFabrication_Wshape.aspx
In the notation Waxb, 'a' denotes the depth in inches and b denotes the weight in pounds per unit length in foot.
Entering these values in the input section and pressing calculate gives the values in the output section.
Before proceeding to calculations, we note the following:
We have to design the beam corresponding to maximum bending stress as it has been mentioned that other loads are negligible. Since there is no axial load, the neutral axis coincides with the centroid of the beam. Also, as the I section is symmetric, the centroid divides the cross section equally.
From the figure we get:
Maximum Bending Moment = M = 17,200 lb-ft.
Yield stress = Y = 36000 psi = 5184000 lb/ft^2. (A36 Structural Steel)
Allowable Maximum Stress = Y/3 = 1728000 lb/ft^2 (Factor of Safety = 3)
Let us do option by option:
a. W6X25
Section Modulus = Z = 16.8 in^3 = 0.0097 ft^3.
Maximum Bending Stress = M/Z = 17,200/0.0097 = 1773195.88 lb/ft^2.
The maximum Bending Stress is more than the allowable load, so this beam will fail under the load. Thus, it is out of consideration.
b. W8X15
Section Modulus = Z = 11.9 in^3 = 0.0069 ft^3.
Maximum Bending Stress = M/Z = 17,200/0.0069 = 2492753.62 lb/ft^2.
The maximum Bending Stress is more than the allowable load, so this beam will fail under the load. Thus, it is out of consideration.
c. W8X24
Section Modulus = Z = 20.9 in^3 = 0.012 ft^3.
Maximum Bending Stress = M/Z = 17,200/0.012 = 1433333.33 lb/ft^2.
The maximum Bending Stress is less than the allowable load, so this beam will sustain the load. Hence we will consider the beam later with other possible candidates.
d. W10X22
Section Modulus = Z = 23.3 in^3 = 0.0135 ft^3.
Maximum Bending Stress = M/Z = 17,200/0.0135 = 1274074.07 lb/ft^2.
The maximum Bending Stress is less than the allowable load, so this beam will sustain the load. Hence we will consider the beam later with other possible candidates.
e. W16X26
Section Modulus = Z = 38.4 in^3 = 0.022 ft^3.
Maximum Bending Stress = M/Z = 17,200/0.022 = 781818.18 lb/ft^2.
The maximum Bending Stress is less than the allowable load, so this beam will sustain the load. Hence we will consider the beam later with other possible candidates.
Thus, the beams which will sustain the load are W8X24, W10X22 and W16X26. Of these beams, the lightest one is W10X22 (according to the notation above). Hence the lightest beam that can support the load is W10X22.
I hope you understood the calculations. Please give a tumbs up!!!
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