Design Strength Calculation Of Flexural Member

In structural engineering, flexural members, such as beams and girders, are essential components that resist bending moments. These members are designed to withstand loads applied perpendicularly to their longitudinal axis, causing them to bend. Determining the design strength of a flexural member is a critical step in ensuring the structural integrity and safety of any construction project. This article will delve 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 dimensions. Understanding the flexural behavior of structural elements is crucial for engineers to design safe and efficient structures. The design strength, a key parameter, represents the member's capacity to resist bending moments without failure. This calculation involves considering various factors, such as the yield strength of the material, the geometry of the cross-section, and the support conditions. This article aims to provide a comprehensive guide to calculating the design strength of a specific flexural member, illustrating the practical application of structural engineering principles. The design strength of a flexural member is its capacity to resist bending moments safely. This calculation involves several factors, including the material's yield strength, the cross-sectional geometry, and the support conditions. The flexural member discussed in this article is fabricated from two flange plates and a web plate, a common configuration in structural steel design. This configuration offers high strength and stiffness, making it suitable for various applications. The design process begins with determining the relevant material properties, such as the yield strength (Fy), which represents the stress at which the material begins to deform permanently. For the given problem, Fy is specified as 248 MPa. Next, the geometric properties of the cross-section, including the dimensions of the flange plates and the web plate, are considered. These dimensions are crucial in calculating the section's moment of inertia, a measure of its resistance to bending. The calculation of design strength also involves checking for various limit states, such as yielding, lateral-torsional buckling, and local buckling. These limit states represent potential failure modes that must be addressed in the design process. By considering these factors, engineers can ensure that the flexural member can safely withstand the applied loads without failure.

Problem Statement

A flexural member is fabricated from two flange plates, each 178 mm x 19 mm thick, and a web plate, 394 mm x 13 mm thick. The steel has a yield strength (Fy) of 248 MPa. Assuming the beam is laterally supported, calculate the design strength of the section in kN-m. This problem highlights the practical application of structural engineering principles in determining the load-bearing capacity of a flexural member. The given dimensions and material properties are essential for calculating the section's geometric properties and assessing its resistance to bending. The condition of lateral support is crucial as it prevents lateral-torsional buckling, a failure mode that can significantly reduce the member's strength. Lateral support is provided by bracing elements that prevent the compression flange from displacing laterally. When a beam is laterally supported, its design strength is primarily governed by its flexural capacity, which is determined by its ability to resist bending moments without yielding or buckling. The calculation involves determining the plastic moment capacity (Mp) of the section, which represents the maximum moment the section can resist when fully yielded. The design strength is then calculated by applying a safety factor to Mp, ensuring that the member can safely withstand the applied loads. The problem statement provides all the necessary information to perform this calculation, including the dimensions of the flange plates and web plate, the yield strength of the steel, and the condition of lateral support. By following the steps outlined in this article, engineers can accurately determine the design strength of the flexural member and ensure its structural integrity. This ensures that the structure can safely carry the intended loads without experiencing any failure due to bending. The design strength is a critical parameter that must be accurately calculated to ensure the safety and reliability of structures.

Material Properties

• Yield Strength (Fy) = 248 MPa

Geometric Properties

• Flange Plates: 178 mm x 19 mm (2 plates)
• Web Plate: 394 mm x 13 mm

The geometric properties of the flexural member are essential for calculating its section modulus and moment of inertia, which are crucial parameters in determining its flexural strength. The flange plates, with dimensions of 178 mm x 19 mm, contribute significantly to the member's resistance to bending due to their location farthest from the neutral axis. The web plate, with dimensions of 394 mm x 13 mm, provides shear resistance and connects the flange plates, ensuring that they act as a single unit. The dimensions of these plates directly influence the section's ability to resist bending moments. To calculate the design strength, it is necessary to first determine the cross-sectional area and the location of the neutral axis. The neutral axis is the axis about which the section bends, and its location is critical for calculating the moment of inertia. For a symmetrical section, the neutral axis is located at the centroid of the section. In this case, the flexural member is symmetrical about its vertical axis, so the neutral axis is located at the mid-depth of the section. The moment of inertia (I) is a measure of the section's resistance to bending and depends on the shape and dimensions of the cross-section. A higher moment of inertia indicates a greater resistance to bending. The moment of inertia is calculated separately for the flange plates and the web plate and then summed to obtain the total moment of inertia of the section. The section modulus (S) is another important geometric property that relates the moment of inertia to the distance from the neutral axis to the extreme fiber of the section. The section modulus is used to calculate the bending stress in the member under a given bending moment. The geometric properties, including the dimensions of the flange plates and web plate, are crucial for calculating the section modulus and moment of inertia, which are essential parameters in determining the flexural strength of the member.

Calculations

1. Calculate the Plastic Section Modulus (Z)

To determine the plastic section modulus (Z), we need to find the plastic neutral axis (PNA). For a symmetrical section, the PNA coincides with the centroidal axis. The plastic section modulus is a geometric property that represents the section's resistance to plastic bending. It is used to calculate the plastic moment capacity (Mp), which is the maximum moment the section can resist when fully yielded. The plastic neutral axis (PNA) is the axis about which the section bends when it reaches its plastic moment capacity. For a symmetrical section, the PNA coincides with the centroidal axis, which simplifies the calculation. The calculation of Z involves dividing the cross-section into compression and tension zones and determining the first moment of area of each zone about the PNA. The plastic section modulus is then the sum of the absolute values of these first moments. For the given flexural member, the cross-section consists of two flange plates and a web plate. The PNA is located at the mid-depth of the section, dividing it into two equal areas. The plastic section modulus is calculated separately for the flange plates and the web plate and then summed to obtain the total plastic section modulus of the section. The plastic section modulus is a critical parameter in determining the design strength of the flexural member. It is used in conjunction with the yield strength (Fy) to calculate the plastic moment capacity (Mp), which represents the maximum moment the section can resist without failure. The plastic section modulus is an important parameter in the design of flexural members, representing the section's resistance to plastic bending and its ability to withstand high bending moments.

a. Flange Plates

• Area of one flange plate (Af) = 178 mm * 19 mm = 3382 mm²
• Distance from the PNA to the centroid of the flange (yf) = (394 mm / 2) + (19 mm / 2) = 206.5 mm
• Plastic moment contribution from both flanges (Zf) = 2 * Af * yf = 2 * 3382 mm² * 206.5 mm = 1396622.6 mm³

b. Web Plate

• Area of the web plate (Aw) = 394 mm * 13 mm = 5122 mm²
• Distance from the PNA to the centroid of the web (yw) = 394 mm / 4 = 98.5 mm
• Plastic moment contribution from the web (Zw) = Aw * yw = 5122 mm² * 98.5 mm = 504517 mm³ Note: Since the web is symmetric about the PNA, we only consider the area above the PNA.

c. Total Plastic Section Modulus

• Z = Zf + Zw = 1396622.6 mm³ + 504517 mm³ = 1901139.6 mm³

The calculation of the plastic section modulus (Z) involves determining the contribution from the flange plates and the web plate, as these are the primary components that resist bending. For the flange plates, the area of each plate (Af) is calculated by multiplying its width and thickness. The distance from the plastic neutral axis (PNA) to the centroid of the flange (yf) is determined by considering the geometry of the section. The plastic moment contribution from both flanges (Zf) is then calculated by multiplying the area of one flange by the distance from the PNA to its centroid and multiplying by 2 (since there are two flanges). For the web plate, the area (Aw) is calculated by multiplying its height and thickness. The distance from the PNA to the centroid of the web (yw) is determined by considering the geometry of the section. The plastic moment contribution from the web (Zw) is then calculated by multiplying the area of the web by the distance from the PNA to its centroid. The total plastic section modulus (Z) is the sum of the plastic moment contributions from the flanges and the web. This value represents the section's ability to resist plastic bending and is a crucial parameter in determining the design strength of the flexural member. The plastic section modulus is calculated by summing the contributions from the flanges and the web, reflecting the composite action of the section in resisting bending.

2. Calculate the Plastic Moment Capacity (Mp)

The plastic moment capacity (Mp) is the maximum moment the section can resist when fully yielded. It is calculated using the formula:

• Mp = Fy * Z
• Mp = 248 MPa * 1901139.6 mm³
• Mp = 471482620.8 N-mm
• Mp = 471.48 kN-m

The plastic moment capacity (Mp) is a critical parameter in structural design, representing the maximum bending moment a section can withstand before undergoing plastic deformation. It is calculated by multiplying the yield strength (Fy) of the material by the plastic section modulus (Z) of the section. The yield strength (Fy) is a material property that indicates the stress at which the material begins to deform permanently. The plastic section modulus (Z) is a geometric property that represents the section's resistance to plastic bending. The plastic moment capacity (Mp) is expressed in units of force times distance, such as N-mm or kN-m. In this case, the yield strength (Fy) is given as 248 MPa, and the plastic section modulus (Z) has been calculated as 1901139.6 mm³. Multiplying these values gives the plastic moment capacity (Mp) in N-mm. To convert the plastic moment capacity (Mp) from N-mm to kN-m, it is divided by 1,000,000. This conversion is necessary to express the result in a more practical unit for structural design calculations. The plastic moment capacity (Mp) is a key factor in determining the design strength of a flexural member. It represents the section's ability to resist bending moments without undergoing plastic deformation, which can lead to structural failure. The plastic moment capacity is used in conjunction with safety factors to determine the allowable bending moment that the member can safely resist.

3. Determine the Design Strength (ΦbMn)

For laterally supported beams, the design flexural strength (ΦbMn) is calculated as:

• Φb = 0.9 (resistance factor for bending)
• Mn = Mp (nominal flexural strength equals plastic moment capacity for laterally supported beams)
• ΦbMn = 0.9 * 471.48 kN-m
• ΦbMn = 424.33 kN-m

The design strength (ΦbMn) is the usable strength of the flexural member after applying a resistance factor (Φb) to the nominal flexural strength (Mn). The resistance factor (Φb) accounts for uncertainties in material properties, fabrication, and loading conditions. For bending, the resistance factor (Φb) is typically 0.9, as specified in structural design codes. The nominal flexural strength (Mn) represents the theoretical capacity of the member to resist bending moments. For laterally supported beams, where lateral-torsional buckling is prevented, the nominal flexural strength (Mn) is equal to the plastic moment capacity (Mp). This is because the member can fully develop its plastic capacity before any buckling occurs. The design strength (ΦbMn) is calculated by multiplying the resistance factor (Φb) by the nominal flexural strength (Mn). This value represents the maximum bending moment the member can safely resist under design loads. The design strength (ΦbMn) is a critical parameter in structural design, ensuring that the member has sufficient capacity to withstand the applied loads with an adequate margin of safety. The design strength is the final value used to assess the safety and adequacy of the flexural member in resisting bending moments.

Conclusion

The design strength of the flexural member, considering the given material and geometric properties and assuming lateral support, is 424.33 kN-m. This calculation demonstrates the process of determining the flexural capacity of a steel member, which is crucial for structural design and ensuring safety. In conclusion, calculating the design strength of a flexural member is a fundamental aspect of structural engineering, ensuring the safety and stability of structures. This article has provided a detailed step-by-step approach to determining the design strength of a flexural member fabricated from flange plates and a web plate, considering the material properties, geometric dimensions, and support conditions. The calculations involved determining the plastic section modulus (Z), the plastic moment capacity (Mp), and the design strength (ΦbMn). The plastic section modulus (Z) represents the section's resistance to plastic bending and is calculated based on the geometry of the cross-section. The plastic moment capacity (Mp) is the maximum moment the section can resist when fully yielded and is calculated by multiplying the yield strength (Fy) of the material by the plastic section modulus (Z). The design strength (ΦbMn) is the usable strength of the flexural member, accounting for uncertainties in material properties, fabrication, and loading conditions. It is calculated by applying a resistance factor (Φb) to the nominal flexural strength (Mn), which, for laterally supported beams, is equal to the plastic moment capacity (Mp). The final design strength of 424.33 kN-m indicates the maximum bending moment the flexural member can safely resist. This value is crucial for structural engineers in designing safe and efficient structures that can withstand the intended loads without failure. By following the steps outlined in this article, engineers can accurately determine the design strength of flexural members and ensure the structural integrity of their designs. The design strength calculation is a critical step in ensuring the safety and reliability of structural elements.