Flexural Member Design Strength Calculation Example

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In structural engineering, flexural members, such as beams and girders, are essential components that resist bending moments and shear forces. These members are designed to withstand applied loads while maintaining structural integrity and stability. Calculating the design strength of a flexural member is a crucial step in ensuring its safe and reliable performance. This article delves into the process of calculating the design strength of a flexural member fabricated from two flange plates and a web plate, considering the material properties and geometric characteristics of the section.

We are tasked with determining the design strength of a flexural member constructed from two flange plates, each measuring 178 mm x 19 mm, and a web plate with dimensions of 394 mm x 13 mm. The member is bent about its strong axis, and the yield strength (Fy) of the material is given as 248 MPa. The beam is assumed to be laterally supported, which simplifies the design calculations by eliminating the consideration of lateral-torsional buckling.

Before proceeding with the design strength calculation, it's essential to gather the necessary material properties and geometric characteristics of the flexural member:

  • Yield Strength (Fy): 248 MPa
  • Flange Plates:
    • Width (bf): 178 mm
    • Thickness (tf): 19 mm
  • Web Plate:
    • Depth (h): 394 mm
    • Thickness (tw): 13 mm

To determine the design strength, we first need to calculate the section properties, including the area (A), moment of inertia (Ix), and plastic modulus (Zx) about the strong axis.

1. Area (A)

The total area of the section is the sum of the areas of the flange plates and the web plate:

A = 2 * (bf * tf) + (h * tw)
A = 2 * (178 mm * 19 mm) + (394 mm * 13 mm)
A = 6762 mm²

2. Moment of Inertia (Ix)

The moment of inertia about the strong axis is calculated using the parallel axis theorem:

Ix = (2 * ((bf * tf³) / 12 + bf * tf * (h/2 + tf/2)²) + (tw * h³) / 12)
Ix = (2 * ((178 mm * (19 mm)³) / 12 + 178 mm * 19 mm * (394 mm/2 + 19 mm/2)²) + (13 mm * (394 mm)³) / 12)
Ix ≈ 3.07 * 10^8 mm⁴

3. Plastic Modulus (Zx)

The plastic modulus is a geometric property that represents the section's resistance to plastic bending. For a symmetrical I-section, the plastic modulus about the strong axis can be calculated as follows:

Zx = (bf * tf * (h + tf)) + (tw * (h/2)²)
Zx = (178 mm * 19 mm * (394 mm + 19 mm)) + (13 mm * (394 mm/2)²)
Zx ≈ 1.34 * 10^6 mm³

The design strength of the flexural member is determined based on the governing limit state, which could be yielding, lateral-torsional buckling, or local buckling. In this case, since the beam is laterally supported, we only need to consider yielding and local buckling. The design strength is the lowest value obtained from these limit states.

1. Yielding Limit State

The design bending moment capacity (Mn) for yielding is calculated as:

Mn = Fy * Zx
Mn = 248 MPa * 1.34 * 10^6 mm³
Mn ≈ 332.32 * 10^6 N-mm
Mn ≈ 332.32 kN-m

The design strength for yielding (ΦbMn) is calculated by applying a resistance factor (Φb) of 0.9:

ΦbMn = 0.9 * Mn
ΦbMn = 0.9 * 332.32 kN-m
ΦbMn ≈ 299.09 kN-m

2. Local Buckling Limit State

Local buckling can occur in the flange or web of the flexural member. To check for local buckling, we need to calculate the width-to-thickness ratios and compare them to the limiting values specified in the relevant design code (e.g., AISC 360). Let's assume we are using the AISC 360 specification.

Flange Local Buckling

The width-to-thickness ratio for the flange (λf) is:

λf = bf / (2 * tf)
λf = 178 mm / (2 * 19 mm)
λf ≈ 4.68

The limiting width-to-thickness ratio for flange local buckling (λpf) is:

λpf = 0.38 * √(E / Fy)

Assuming E (modulus of elasticity) = 200,000 MPa:

λpf = 0.38 * √(200,000 MPa / 248 MPa)
λpf ≈ 10.79

Since λf (4.68) < λpf (10.79), the flange is not susceptible to local buckling.

Web Local Buckling

The width-to-thickness ratio for the web (λw) is:

λw = h / tw
λw = 394 mm / 13 mm
λw ≈ 30.31

The limiting width-to-thickness ratio for web local buckling (λpw) is:

λpw = 3.76 * √(E / Fy)

Assuming E (modulus of elasticity) = 200,000 MPa:

λpw = 3.76 * √(200,000 MPa / 248 MPa)
λpw ≈ 106.99

Since λw (30.31) < λpw (106.99), the web is not susceptible to local buckling under yielding limit state. However, we also need to check for web local buckling under inelastic conditions. The limiting width-to-thickness ratio for web local buckling (λrw) is:

λrw = 5.70 * √(E / Fy)
λrw = 5.70 * √(200,000 MPa / 248 MPa)
λrw ≈ 162.17

Since λw (30.31) < λrw (162.17), the web is not susceptible to local buckling under inelastic conditions as well. Therefore, local buckling does not govern the design strength.

3. Design Strength

Since yielding governs the design strength and local buckling is not a concern, the design strength of the flexural member is:

Design Strength = ΦbMn ≈ 299.09 kN-m

The design strength of the flexural member, fabricated from two flange plates (178 mm x 19 mm) and a web plate (394 mm x 13 mm) with a yield strength of 248 MPa, is approximately 299.09 kN-m. This calculation considers the yielding limit state and confirms that local buckling is not a governing factor due to the section's geometry and material properties. It is crucial to ensure that the applied bending moment does not exceed this design strength to maintain the structural integrity and safety of the flexural member. These calculations are essential for structural engineers in designing safe and reliable structures.

It is important to note that this calculation assumes the beam is laterally supported, preventing lateral-torsional buckling. In cases where lateral support is not provided, additional calculations considering lateral-torsional buckling would be necessary. Furthermore, adherence to relevant design codes and standards, such as AISC 360, is essential for accurate and safe structural design.

The process of determining the design strength of a flexural member involves careful consideration of material properties, geometric characteristics, and potential failure modes. This comprehensive approach ensures the structural integrity and safety of the designed member under applied loads. This detailed calculation underscores the importance of precise engineering analysis in structural design.

The design strength calculation presented here provides a clear understanding of the steps involved in evaluating the load-carrying capacity of a flexural member. By considering the yielding limit state and local buckling, engineers can ensure that structures are designed to withstand applied loads safely and reliably. The use of appropriate design codes and standards is crucial in ensuring the accuracy and reliability of these calculations.

  • Flexural Member Design Strength Calculation
  • Yield Strength of Steel
  • Moment of Inertia Calculation
  • Plastic Modulus Calculation
  • Local Buckling Check
  • Laterally Supported Beam Design
  • AISC 360 Design Standards
  • Structural Engineering Design
  • Beam Design Strength
  • Steel Beam Capacity

1. What is the design strength of a flexural member?

The design strength of a flexural member is the maximum bending moment that the member can resist without failure. It is a critical parameter in structural design, ensuring the safety and stability of structures under load. This strength is calculated based on material properties, section geometry, and relevant design codes such as AISC 360. Understanding design strength is essential for engineers to create safe and efficient structures. Accurately determining design strength prevents structural failures and ensures the longevity of the structure.

2. How do you calculate the design strength of a flexural member?

To calculate the design strength of a flexural member, several steps are involved. First, determine the material properties, such as yield strength (Fy) and modulus of elasticity (E). Then, calculate the section properties, including area (A), moment of inertia (Ix), and plastic modulus (Zx). Next, assess the limit states, such as yielding and local buckling. For yielding, the bending moment capacity (Mn) is calculated as Fy * Zx, and the design strength is typically 0.9 * Mn (using a resistance factor of 0.9). Local buckling checks involve comparing width-to-thickness ratios to limiting values specified in design codes. The lowest value obtained from these limit states governs the design strength. Accurate calculation of design strength is crucial for structural safety. The process ensures that the member can withstand the applied loads without failure. Proper calculation of the design strength also contributes to the efficient use of materials in construction.

3. What factors affect the design strength of a flexural member?

Several factors influence the design strength of a flexural member. Material properties, such as yield strength (Fy) and modulus of elasticity (E), play a significant role. Geometric properties, including the section's dimensions and shape (e.g., width, thickness, and depth), also affect the design strength. The presence of lateral supports impacts the consideration of lateral-torsional buckling. Limit states, such as yielding and local buckling (flange and web), are crucial considerations. Additionally, design codes and standards (e.g., AISC 360) provide guidelines and limiting values that affect the calculation of design strength. All these factors must be carefully evaluated to ensure an accurate design strength calculation. Variations in these factors can significantly alter the load-carrying capacity of the member. Therefore, a thorough understanding of these factors is essential for structural engineers.

4. What is the role of lateral support in the design strength of a flexural member?

Lateral support plays a crucial role in the design strength of a flexural member. When a flexural member is laterally supported, it prevents or reduces the risk of lateral-torsional buckling, a failure mode where the member twists and deflects sideways under load. With adequate lateral support, the design strength calculation primarily focuses on yielding and local buckling. Without lateral support, the design strength must also account for lateral-torsional buckling, which can significantly reduce the member's capacity. The spacing and effectiveness of lateral supports are critical design considerations. Properly designed lateral supports ensure that the flexural member can achieve its full bending capacity. The presence of lateral supports simplifies the design process and often results in a more efficient use of materials.

5. How do local buckling checks affect the design strength of a flexural member?

Local buckling checks are essential in determining the design strength of a flexural member. Local buckling refers to the buckling of individual components of the section, such as the flange or web, before the entire member reaches its yielding capacity. These checks involve comparing the width-to-thickness ratios of the flange and web to limiting values specified in design codes (e.g., AISC 360). If these ratios exceed the limits, the member is susceptible to local buckling, which reduces its design strength. Stiffeners or adjustments to the section's geometry may be necessary to prevent local buckling. Accurate local buckling checks ensure that the member can sustain loads without premature failure. The outcome of these checks directly influences the design strength calculation and the overall safety of the structure. Addressing local buckling effectively is a key aspect of structural design.