Understanding the Calculation of Axial Force in Structural Elements

Axial force calculation determines the internal force acting along a member’s longitudinal axis. This article explores detailed methods and applications of axial force computation.

Readers will find comprehensive formulas, variable explanations, common values, and real-world examples for precise axial force analysis.

• Calculate axial force in a steel rod subjected to tensile load of 50 kN.
• Determine axial force in a concrete column supporting a 200 kN compressive load.
• Find axial force in a truss member with combined axial and bending loads.
• Compute axial force in a pipeline segment under internal pressure and external loads.

Comprehensive Tables of Common Axial Force Values

Fundamental Formulas for Axial Force Calculation

Axial force (F) is the internal force transmitted along the longitudinal axis of a structural member. It can be tensile or compressive depending on the nature of the applied load.

Basic Axial Force Formula

F = P

Where:

• F = Axial force (kN or N)
• P = Applied axial load (kN or N)

This formula assumes the load is purely axial without bending or shear components.

Axial Stress Calculation

Axial stress (σ) is the normal stress induced by axial force and is calculated as:

σ = F / A

Where:

• σ = Axial stress (MPa or N/mm²)
• F = Axial force (N)
• A = Cross-sectional area (mm²)

Typical values for A depend on the member geometry and material specifications.

Axial Force in Combined Loading

When axial force is combined with bending moments, the total normal stress at a point is:

σ_total = (F / A) ± (M * c) / I

Where:

• M = Bending moment (N·mm)
• c = Distance from neutral axis to outer fiber (mm)
• I = Moment of inertia of cross-section (mm⁴)

The sign depends on whether the bending stress is tensile or compressive at the point considered.

Axial Force from Internal Pressure in Cylindrical Members

For pipes or pressure vessels, axial force due to internal pressure (p) is:

F = p * A

Where:

• p = Internal pressure (Pa or N/m²)
• A = Cross-sectional area resisting the pressure (m²)

For thin-walled cylinders, the axial force can also be related to hoop stress and longitudinal stress as per ASME Boiler and Pressure Vessel Code.

Allowable Axial Load Based on Material Strength

To ensure safety, the allowable axial load (P_allow) is calculated as:

P_allow = A * σ_allow

Where:

• σ_allow = Allowable stress of the material (MPa)
• A = Cross-sectional area (mm²)

Allowable stress is typically a fraction of the material’s yield strength, incorporating safety factors as per design codes such as AISC or Eurocode.

Detailed Explanation of Variables and Common Values

• Axial Force (F): The internal force along the member’s axis, measured in Newtons (N) or kiloNewtons (kN). It can be tensile (pulling) or compressive (pushing).
• Applied Load (P): External load applied to the member, often given in kN or N. This is the primary input for axial force calculation.
• Cross-Sectional Area (A): The area of the member’s cross-section perpendicular to the axial force, measured in mm² or m². Common shapes include circular, rectangular, and I-beams.
• Axial Stress (σ): Stress induced by axial force, calculated as force divided by area, measured in MPa (N/mm²). Typical allowable stresses vary by material.
• Bending Moment (M): Moment causing bending, measured in N·mm or kN·m. Important when axial force is combined with bending.
• Distance to Outer Fiber (c): Distance from neutral axis to the extreme fiber of the cross-section, in mm.
• Moment of Inertia (I): Geometric property of the cross-section indicating resistance to bending, in mm⁴.
• Internal Pressure (p): Pressure inside cylindrical members, in Pascals (Pa) or N/m².
• Allowable Stress (σ_allow): Maximum permissible stress for design, factoring in safety margins.

Real-World Applications and Case Studies

Case Study 1: Axial Force in a Steel Column Supporting a Multi-Storey Building

A steel column with a cross-sectional area of 1200 mm² supports a compressive load of 400 kN from the floors above. The steel grade is A992 with a yield strength of 450 MPa. Determine the axial stress and verify if the column is safe under the applied load.

Step 1: Calculate axial stress

σ = F / A = 400,000 N / 1200 mm² = 333.33 MPa

Step 2: Determine allowable stress

Assuming a safety factor of 1.5, allowable stress is:

σ_allow = 450 MPa / 1.5 = 300 MPa

Step 3: Compare axial stress with allowable stress

Since 333.33 MPa > 300 MPa, the column is overstressed and requires redesign, either by increasing cross-sectional area or using higher strength steel.

Case Study 2: Axial Force in a Concrete Column with Combined Axial and Bending Loads

A reinforced concrete column with a rectangular cross-section 300 mm x 500 mm supports an axial compressive load of 600 kN and a bending moment of 50 kN·m. Calculate the maximum compressive stress at the extreme fiber.

Step 1: Calculate cross-sectional area

A = 300 mm * 500 mm = 150,000 mm²

Step 2: Calculate axial stress

σ_axial = F / A = 600,000 N / 150,000 mm² = 4 MPa

Step 3: Calculate moment of inertia (I) for rectangular section

I = (b * h³) / 12 = (300 mm * (500 mm)³) / 12 = 3.125 × 10^9 mm⁴

Step 4: Calculate distance to extreme fiber (c)

c = h / 2 = 500 mm / 2 = 250 mm

Step 5: Calculate bending stress

σ_bending = (M * c) / I = (50,000 N·m * 250 mm) / 3.125 × 10^9 mm⁴

Convert moment to N·mm:

50,000 N·m = 50,000,000 N·mm

Then:

σ_bending = (50,000,000 N·mm * 250 mm) / 3.125 × 10^9 mm⁴ = 4 MPa

Step 6: Calculate total maximum compressive stress

σ_total = σ_axial + σ_bending = 4 MPa + 4 MPa = 8 MPa

This stress is well within typical allowable compressive stress for concrete (~20-30 MPa), indicating the column is safe under combined loading.

Additional Considerations in Axial Force Calculations

• Load Eccentricity: Real-world loads often do not act exactly through the centroid, causing bending moments that must be included in stress calculations.
• Material Nonlinearity: For materials like concrete, nonlinear stress-strain behavior affects axial force capacity and must be considered in advanced analysis.
• Buckling Effects: Compressive axial forces can cause buckling in slender members, requiring stability analysis per Euler’s formula or design codes.
• Temperature Effects: Thermal expansion or contraction can induce axial forces in restrained members.
• Dynamic Loads: Axial forces from impact or cyclic loading require fatigue and dynamic analysis.

Authoritative References and Further Reading

• American Institute of Steel Construction (AISC) Steel Construction Manual
• Eurocode 3: Design of Steel Structures
• ASME Boiler and Pressure Vessel Code
• FHWA Engineering Circular on Axial Load Effects
• American Concrete Institute (ACI) Codes and Standards

Mastering axial force calculation is essential for structural integrity and safety. This article provides the technical foundation and practical tools for engineers and designers to perform accurate axial force analysis across diverse applications.