To compute the nominal strength based on the net section for the tension member, we follow these steps:

Given Data:

• Plate size: $\frac{1}{2} \times 3 \times 10$ plate of A36 steel.
• Bolt diameter: $7/8$ inch.
• Material: A36 steel.
• The connection is bolted, meaning we will use the net section area and the tensile strength to find the nominal strength.

1. Net Area Calculation:

The net area $A_n$ is calculated by subtracting the area lost due to the holes from the gross area. We use the following equation:

$A_n = A_g - n \cdot d_h \cdot t$

Where:

• $A_n$ = Net area.
• $A_g$ = Gross area of the plate.
• $n$ = Number of bolt holes along the line of stress.
• $d_h$ = Hole diameter (this includes a 1/16 inch allowance for bolt clearance, i.e., $d_h = d_b + 1/16$).
• $t$ = Plate thickness.

a) Gross Area $A_g$:

The gross area is the product of the plate width and thickness:

$A_g = 3 \, \text{in} \times \frac{1}{2} \, \text{in} = 1.5 \, \text{in}^2$

b) Hole Diameter $d_h$:

For the $7/8$-inch diameter bolts, the hole diameter $d_h$ includes an allowance of $1/16$-inch:

$d_h = 7/8 \, \text{in} + 1/16 \, \text{in} = 0.875 \, \text{in} + 0.0625 \, \text{in} = 0.9375 \, \text{in}$

c) Net Area $A_n$:

If there is one bolt hole along the critical section (the path of failure), we subtract the area lost due to that hole:

$A_n = 1.5 \, \text{in}^2 - 1 \times 0.9375 \, \text{in} \times \frac{1}{2} \, \text{in}$ $A_n = 1.5 \, \text{in}^2 - 0.46875 \, \text{in}^2 = 1.03125 \, \text{in}^2$

2. Nominal Tensile Strength Calculation:

The nominal tensile strength $P_n$ based on the net section is given by:

$P_n = F_u \times A_n$

Where:

• $F_u$ = Ultimate tensile strength of A36 steel = 58 ksi = 58,000 psi.
• $A_n$ = Net area calculated above.

Substituting the values:

$P_n = 58,000 \, \text{psi} \times 1.03125 \, \text{in}^2$ $P_n = 59,812.5 \, \text{lb} = 59.81 \, \text{kips}$

Conclusion:

The nominal strength of the tension member based on the net section is 59.81 kips.

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Would you like to go into more details or have any questions?

Here are some related questions for further study:

1. How does the number of bolt holes affect the net section strength?
2. What is the role of ultimate tensile strength in determining the nominal strength?
3. How is the gross area different from the net area in bolted connections?
4. What are the safety factors applied in the design based on the nominal strength?
5. How would increasing the thickness of the plate influence the nominal strength?

Tip: Always remember to add clearance (typically $1/16$ inch) to the bolt diameter when calculating hole diameters.