Let's solve **Problem #1** step-by-step: --- ### **Given Data** - **Effective depth of beam, \( d \) = 600 mm** - **Web thickness, \( b_w \) = 300 mm** - **Spacing of beam, \( s \) = 2.4 m = 2400 mm** - **Thickness of slab, \( h \) = 100 mm** - **Span of beam, \( L \) = 6 m = 6000 mm** --- ### **Step 1: Formula for Effective Flange Width (\( b_f \))** For an interior L-beam, the effective flange width (\( b_f \)) is the **least** of the following: 1. \(\frac{L}{6} + b_w + 6h\) 2. Spacing of beams (center to center) 3. \(b_w + \frac{1}{2} \times \) (sum of distances to adjacent beams) For typical interior beams in slab systems, the most common formula is: \[ b_f = \min\left[ b_w + \frac{L}{6},\ s \right] \] However, as per ACI code, for a T-beam: \[ b_f = \min\left[ b_w + \frac{1}{2}l_0, \ s \right] \] where \( l_0 \) is the span of the beam. But for the **end (exterior) L-beam**: \[ b_f = \min\left[ b_w + \frac{1}{12}L + 3h,\ s \right] \] But since the problem asks for the **end of the beam or L-beam**, and based on common practice for interior beams: \[ b_f = \min\left[ b_w + \frac{L}{6} + 6h,\ s \right] \] --- ### **Step 2: Substitute the Values** #### 1. \(\frac{L}{6} + b_w + 6h\): - \( \frac{L}{6} = \frac{6000\ \text{mm}}{6} = 1000\ \text{mm} \) - \( 6h = 6 \times 100\ \text{mm} = 600\ \text{mm} \) - So, \[ b_f = b_w + \frac{L}{6} + 6h = 300\ \text{mm} + 1000\ \text{mm} + 600\ \text{mm} = 1900\ \text{mm} \] #### 2. Spacing of beams: - \( s = 2400\ \text{mm} \) --- ### **Step 3: Take the Lesser Value** \[ b_f = \min(1900\ \text{mm},\ 2400\ \text{mm}) \] So, \[ \boxed{b_f = 1900\ \text{mm}} \] --- ### **Final Answer** **The effective width of the flange of the end of the beam (L-beam) is** \[ \boxed{1900\ \text{mm}} \] --- **Summary Table:** | Calculation | Value | |------------------------------|-----------| | \( b_w + \frac{L}{6} + 6h \) | 1900 mm | | Spacing \( s \) | 2400 mm | | **Effective width (\( b_f \))** | **1900 mm** | --- **Explanation:** We use the code formula for effective flange width, substitute the given values, and take the minimum as the answer.

Certainly! Here's the detailed solution divided into three parts, each with its explanation: --- ### **Part 1: Understanding the Data and Relevant Formula** **Given Data:** - Effective depth of beam \( d = 600 \) mm - Web thickness \( b_w = 300 \) mm - Spacing between beams \( s = 2.4 \) m = 2400 mm - Slab thickness \( h = 100 \) mm - Span of the beam \( L = 6 \) m = 6000 mm **Relevant Concept:** For an **end (exterior) L-beam**, the *effective flange width* \( b_f \) is generally determined by the minimum of two values: 1. The sum of the web width plus contributions from the slab and the span, usually expressed as: \[ b_w + \frac{L}{6} + 6h \] 2. The spacing between beams \( s \). This formula ensures the effective flange width accounts for the beam's load-carrying capacity and shared support from adjacent elements. --- **Explanation:** Understanding the problem's parameters and the applicable code formula helps us determine the effective flange width. The key is recognizing that the flange width can't be arbitrarily large; it's limited by the beam spacing and the contributions from the slab and span. --- ### **Part 2: Calculating the Two Candidate Values** **First candidate:** \[ b_w + \frac{L}{6} + 6h \] Calculations: - \( \frac{L}{6} = \frac{6000}{6} = 1000 \) mm - \( 6h = 6 \times 100 = 600 \) mm Adding these with \( b_w \): \[ 300 + 1000 + 600 = 1900 \text{ mm} \] **Second candidate:** \[ s = 2400 \text{ mm} \] **Comparison:** - First value: 1900 mm - Second value: 2400 mm --- **Explanation:** We compute both possible effective flange widths based on the formula. The first accounts for the beam's span, slab thickness, and web width, while the second is simply the spacing between beams. The effective flange width must be the smaller of these two to ensure safety and compliance with design standards. --- ### **Part 3: Finalizing the Effective Flange Width** **Determine the minimum:** \[ b_f = \min(1900 \text{ mm}, 2400 \text{ mm}) = 1900 \text{ mm} \] **Final Answer:** \[ \boxed{\text{Effective flange width} = 1900 \text{ mm}} \] --- **Explanation:** Since the effective flange width cannot exceed the lesser of the two calculated values, the final effective width is 1900 mm. This ensures the flange width is appropriate for supporting the slab and load distribution near the end of the beam while respecting the spacing constraints. --- **Summary:** - Part 1: Understand the data and the formula. - Part 2: Calculate the two candidate widths. - Part 3: Choose the minimum as the effective flange width. This structured approach guarantees clarity and accuracy in the design process.

Certainly! Here's the detailed solution divided into **two parts**, each with its own explanation: --- ### **Part 1: Understanding Data and Calculating the First Candidate Width** **Given Data:** - Effective depth of beam, \( d = 600 \) mm - Web thickness, \( b_w = 300 \) mm - Spacing between beams, \( s = 2400 \) mm - Slab thickness, \( h = 100 \) mm - Span of the beam, \( L = 6000 \) mm **Relevant Formula for Effective Flange Width (\( b_f \)) (for end/l-beam):** \[ b_f = \min \left[ b_w + \frac{L}{6} + 6h,\ s \right] \] **Step 1: Calculate \( b_w + \frac{L}{6} + 6h \)** - \( \frac{L}{6} = \frac{6000}{6} = 1000 \) mm - \( 6h = 6 \times 100 = 600 \) mm Adding these: \[ b_w + \frac{L}{6} + 6h = 300 + 1000 + 600 = 1900\ \text{mm} \] --- **Explanation:** In this part, we interpret the problem data and apply the standard code formula for the effective flange width of an end or L-beam. Calculating this sum gives us one candidate value for the flange width, which considers the beam's span, slab thickness, and web width. --- ### **Part 2: Comparing and Finalizing the Effective Flange Width** **Step 2: Recall the beam spacing:** \[ s = 2400\ \text{mm} \] **Step 3: Determine the effective flange width:** \[ b_f = \min(1900\ \text{mm},\ 2400\ \text{mm}) = 1900\ \text{mm} \] --- **Final Answer:** \[ \boxed{\text{Effective flange width} = 1900\ \text{mm}} \] --- **Explanation:** In this final part, we compare the two candidate widths: 1900 mm (from the calculation considering span and slab thickness) and 2400 mm (beam spacing). Since the effective flange width cannot be larger than the spacing between beams, we select the minimum value, which is 1900 mm. This ensures the design adheres to safety and code standards, providing a proper effective flange width at the end of the beam. --- **Summary:** - Part 1: Calculate the first candidate width based on span and slab parameters. - Part 2: Compare with the beam spacing and select the minimum as the effective flange width. This structured approach ensures clarity and correctness in the design process.