Let's break down the solution step by step for the provided **built-up section**. We'll need to calculate: ### (i) **Area in sq.mm** Let’s identify and sum up the areas of all rectangles (outer and inner voids). #### **Step 1: Outer Rectangle** - Height = 617.85 mm - Width = 445.58 mm - Area_outer = 617.85 mm × 445.58 mm = **275,228.01 mm²** #### **Step 2: Subtract Inner Voids** There are two major inner voids (central rectangles): - **Void 1:** 245.58 mm × 245.58 mm - **Void 2:** 235.58 mm × 7.58 mm - Area_void1 = 245.58 × 245.58 = **60,329.94 mm²** - Area_void2 = 235.58 × 7.58 = **1,786.77 mm²** #### **Step 3: Subtract Other Inner Areas** - Right vertical void: 8.47 mm × 245.58 mm = **2,080.08 mm²** - Top void: 10.85 mm × 245.58 mm = **2,664.05 mm²** - Left void: 10.85 mm × 7.58 mm = **82.21 mm²** #### **Step 4: Total Area** Area_total = Area_outer - (Area_void1 + Area_void2 + Area_right + Area_top + Area_left) Area_total = 275,228.01 - (60,329.94 + 1,786.77 + 2,080.08 + 2,664.05 + 82.21) Area_total = 275,228.01 - 66,942.95 Area_total = **208,285.06 mm²** --- ### (ii) **Ix, Iy about the given axis (in ×10⁶ mm⁴)** This requires finding the **moment of inertia** for the built-up section. For a composite area: #### For a rectangle about its centroid: - \( I_x = \frac{bh^3}{12} \) - \( I_y = \frac{hb^3}{12} \) ##### 1. Outer Rectangle: - \( I_{x, \text{outer}} = \frac{445.58 \times (617.85)^3}{12} = \frac{445.58 \times 236,040,211.6}{12} = 8,764,491,032.6 \) mm⁴ - \( I_{y, \text{outer}} = \frac{617.85 \times (445.58)^3}{12} = \frac{617.85 \times 88,455,759.2}{12} = 4,558,843,916.5 \) mm⁴ ##### 2. Subtract inner rectangles/voids (use parallel axis theorem if not centered): Repeat calculation for each void and subtract from the outer rectangle's values. For simplicity, let's estimate the **Ix** and **Iy** as: - Ix ≈ 8,764,491,032.6 - (Ix_voids) - Iy ≈ 4,558,843,916.5 - (Iy_voids) Let’s estimate the major void (central): - \( I_{x, \text{void1}} = \frac{245.58 \times (245.58)^3}{12} = \frac{245.58 \times 14,819,992.6}{12} = 303,277,528.7 \) mm⁴ - \( I_{y, \text{void1}} = \frac{245.58 \times (245.58)^3}{12} = 303,277,528.7 \) mm⁴ Do this for all voids and subtract from the outer. #### **Convert to ×10⁶ mm⁴:** - Ix ≈ (Result above) / 1,000,000 - Iy ≈ (Result above) / 1,000,000 --- ### (iii) **Section Modulus (in ×10³ mm³)** Section modulus \( S = \frac{I}{c} \), where \( c \) is the distance from the centroid to the outermost fiber. - For x-axis (vertical): \( c = \frac{617.85}{2} = 308.93 \) mm - For y-axis (horizontal): \( c = \frac{445.58}{2} = 222.79 \) mm So, - \( S_x = \frac{I_x}{c_x} \) - \( S_y = \frac{I_y}{c_y} \) --- ### (iv) **Radius of Gyration (in mm)** \( r = \sqrt{\frac{I}{A}} \) Do this for both axes: - \( r_x = \sqrt{\frac{I_x}{A}} \) - \( r_y = \sqrt{\frac{I_y}{A}} \) --- ## **Summary Table (Fill in after calculations):** | Property | Value (approx.) | |--------------------|-----------------------| | Area (mm²) | 208,285 | | Ix (×10⁶ mm⁴) | [calculate] | | Iy (×10⁶ mm⁴) | [calculate] | | Sx (×10³ mm³) | [calculate] | | Sy (×10³ mm³) | [calculate] | | rx, ry (mm) | [calculate] | --- ### **Note:** - For exact inertia and section modulus, sum/subtract all rectangles and voids using the parallel axis theorem as necessary. - If you need the full step-by-step calculation for each rectangle/void, let me know! Would you like the full detailed calculation for moments of inertia, section modulus, etc. for each component?

Certainly! Let's perform detailed step-by-step calculations for the given built-up section. --- ## Step 1: **Calculate the Total Area (A)** ### Outer Rectangle: - Height \( h_{outer} = 617.85\, \text{mm} \) - Width \( b_{outer} = 445.58\, \text{mm} \) \[ A_{outer} = 617.85 \times 445.58 = \boxed{275,228.01\, \text{mm}^2} \] ### Void 1 (central rectangle): - Dimensions: \( 245.58\, \text{mm} \times 245.58\, \text{mm} \) \[ A_{void1} = 245.58 \times 245.58 = (245.58)^2 = \boxed{60,329.94\, \text{mm}^2} \] ### Void 2 (horizontal slit): - Dimensions: \( 7.58\, \text{mm} \times 235.58\, \text{mm} \) \[ A_{void2} = 7.58 \times 235.58 = \boxed{1,786.77\, \text{mm}^2} \] ### Other internal voids: - **Right vertical void**: \( 8.47\, \text{mm} \times 245.58\, \text{mm} \) \[ A_{right} = 8.47 \times 245.58 = \boxed{2,080.08\, \text{mm}^2} \] - **Top void**: \( 10.85\, \text{mm} \times 245.58\, \text{mm} \) \[ A_{top} = 10.85 \times 245.58 = \boxed{2,664.05\, \text{mm}^2} \] - **Left vertical void**: \( 10.85\, \text{mm} \times 7.58\, \text{mm} \) \[ A_{left} = 10.85 \times 7.58 = \boxed{82.21\, \text{mm}^2} \] ### Total net area: \[ A_{total} = A_{outer} - (A_{void1} + A_{void2} + A_{right} + A_{top} + A_{left}) \] \[ A_{total} = 275,228.01 - (60,329.94 + 1,786.77 + 2,080.08 + 2,664.05 + 82.21) = 275,228.01 - 66,942.95 = \boxed{208,285.06\, \text{mm}^2} \] --- ## Step 2: **Calculate Moments of Inertia (Ix and Iy)** ### Step 2.1: Moment of Inertia for Outer Rectangle about its centroid: \[ I_{x, outer} = \frac{b_{outer} \times h_{outer}^3}{12} \] \[ I_{x, outer} = \frac{445.58 \times (617.85)^3}{12} \] Calculate \( (617.85)^3 \): \[ 617.85^3 = 617.85 \times 617.85 \times 617.85 \approx 236,040,211.6\, \text{mm}^3 \] Now, \[ I_{x, outer} = \frac{445.58 \times 236,040,211.6}{12} = \frac{105,096,174,464}{12} \approx 8.758 \times 10^{9}\, \text{mm}^4 \] Similarly, for \( I_{y, outer} \): \[ I_{y, outer} = \frac{h_{outer} \times b_{outer}^3}{12} \] Calculate \( (445.58)^3 \): \[ 445.58^3 \approx 88,455,759.2\, \text{mm}^3 \] \[ I_{y, outer} = \frac{617.85 \times 88,455,759.2}{12} \approx \frac{54,673,278,537}{12} \approx 4.556 \times 10^{9}\, \text{mm}^4 \] --- ### Step 2.2: Moment of Inertia for the voids Since the voids are subtracted, their moments of inertia are also considered, using the parallel axis theorem if their centroid is not at the neutral axis. --- ### Step 2.3: Approximate moments for the voids: Assuming the voids are centered at the centroid of the outer rectangle and aligned symmetrically, their moments of inertia about the neutral axes: - **Void 1 (245.58 × 245.58):** \[ I_{x, void1} = \frac{b_{void1} \times h_{void1}^3}{12} = \frac{245.58 \times (245.58)^3}{12} \] Calculate: \[ (245.58)^3 = 14,819,992.6\, \text{mm}^3 \] \[ I_{x, void1} = \frac{245.58 \times 14,819,992.6}{12} \approx \frac{3,636,346,308}{12} \approx 303,029,692\, \text{mm}^4 \] Similarly, \( I_{y, void1} \) is identical due to symmetry: \[ I_{y, void1} = 303,029,692\, \text{mm}^4 \] - **Void 2 (7.58 × 235.58):** \[ I_{x, void2} = \frac{7.58 \times (235.58)^3}{12} \] Calculate: \[ (235.58)^3 = 13,144,126.7\, \text{mm}^3 \] \[ I_{x, void2} = \frac{7.58 \times 13,144,126.7}{12} \approx \frac{99,565,604}{12} \approx 8,297,133.6\, \text{mm}^4 \] Similarly, \( I_{y, void2} \): \[ I_{y, void2} = \frac{235.58 \times 7.58^3}{12} \] Calculate \( 7.58^3 \): \[ 7.58^3 \approx 435.2 \] \[ I_{y, void2} = \frac{235.58 \times 435.2}{12} \approx \frac{102,481.4}{12} \approx 8,540.1\, \text{mm}^4 \] --- ### Step 2.4: Parallel Axis Theorem adjustments Since the voids are located at certain distances from the centroid, we need to add the term: \[ I_{parallel} = A_{void} \times d^2 \] where \( d \) is the distance between the centroid of the void and the neutral axis. Assuming the voids are symmetrically placed, their centroid distances for each are: - For the central void (void 1): \[ d_{void1} \approx 0\, \text{mm} \quad (\text{centered at neutral axis}) \] - For the horizontal slit (void 2): Assuming it is located at a certain height or width; for simplicity, let's estimate the distances as zero given symmetry. **Note:** For precise calculation, their positions relative to the centroid are needed. Given the symmetry, the centroidal distances are zero or minimal. --- ## **Final Moments of Inertia:** \[ I_{x} \approx I_{x, outer} - I_{x, void1} - I_{x, void2} \] \[ I_{x} \approx 8.758 \times 10^{9} - 0.303 \times 10^{9} - 0.0083 \times 10^{9} = 8.4467 \times 10^{9}\, \text{mm}^4 \] Similarly, \[ I_{y} \approx 4.556 \times 10^{9} - 0.303 \times 10^{9} - 0.0085 \times 10^{9} = 4.244 \times 10^{9}\, \text{mm}^4 \] --- ## Step 3: **Section Modulus** Calculate the distance from centroid to outermost fiber: - For **x-axis (vertical bending)**: \[ c_x = \frac{h_{outer}}{2} = \frac{617.85}{2} = 308.93\, \text{mm} \] - For **y-axis (horizontal bending)**: \[ c_y = \frac{b_{outer}}{2} = \frac{445.58}{2} = 222.79\, \text{mm} \] Section modulus: \[ S_x = \frac{I_x}{c_x} = \frac{8.447 \times 10^{9}}{308.93} \approx 27.36 \times 10^{6}\, \text{mm}^3 \] \[ S_y = \frac{I_y}{c_y} = \frac{4.244 \times 10^{9}}{222.79} \approx 19.05 \times 10^{6}\, \text{mm}^3 \] Expressed in ×10³ mm³: \[ S_x \approx 27360 \times 10^{3}\, \text{mm}^3 \] \[ S_y \approx 19050 \times 10^{3}\, \text{mm}^3 \] --- ## Step 4: **Radius of Gyration** \[ r_x = \sqrt{\frac{I_x}{A}} = \sqrt{\frac{8.447 \times 10^{9}}{208,285}} \approx \sqrt{40,526} \approx 201.3\, \text{mm} \] \[ r_y = \sqrt{\frac{I_y}{A}} = \sqrt{\frac{4.244 \times 10^{9}}{208,285}} \approx \sqrt{20,263} \approx 142.4\, \text{mm} \] --- ## **Final Summary:** | Property | Value | |------------------------------|-------------------------------------| | **Area (mm²)** | **208,285** | | **Moment of Inertia Ix (×10⁶ mm⁴)** | **8.447** | | **Moment of Inertia Iy (×10⁶ mm⁴)** | **4.244** | | **Section Modulus Sx (×10³ mm³)** | **27,360** | | **Section Modulus Sy (×10³ mm³)** | **19,050** | | **Radius of Gyration rx (mm)** | **201.3** | | **Radius of Gyration ry (mm)** | **142.4** | --- **Note:** These calculations are based on symmetry assumptions and approximate positions; for exact results, detailed coordinate positions of each void are necessary. --- If you'd like, I can assist further with precise positional calculations or any other property!