Understanding the Calculation of Shear Force (by Cutting)

Shear force calculation by cutting is essential in structural and mechanical engineering. It determines internal forces resisting sliding failure.

This article explores detailed formulas, common values, and real-world applications of shear force calculation by cutting.

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Calculate shear force for a simply supported beam with a central point load.
Determine shear force in a steel rod subjected to torsion and axial load.
Find shear force in a rectangular beam with distributed load and varying cross-section.
Compute shear force in a welded joint under combined bending and shear stresses.

Comprehensive Tables of Common Values in Shear Force Calculation

Material	Shear Strength (τ_max) (MPa)	Typical Cross-Sectional Area (A) (mm²)	Common Load Types	Typical Shear Force Range (N)
Structural Steel (A36)	250	1000 – 5000	Point Load, Distributed Load	10,000 – 1,250,000
Aluminum Alloy (6061-T6)	170	500 – 3000	Shear in Bolted Joints	5,000 – 510,000
Concrete (Normal Strength)	3 – 5	10,000 – 50,000	Shear in Beams and Slabs	30,000 – 250,000
Timber (Douglas Fir)	5 – 10	2000 – 8000	Shear in Joists and Rafters	10,000 – 80,000
High Strength Bolts (Grade 8.8)	400	50 – 200	Shear in Fasteners	20,000 – 80,000
Reinforcing Steel (Rebar)	250	150 – 1000	Shear in Reinforced Concrete	37,500 – 250,000
Cast Iron	100	1000 – 4000	Shear in Machine Components	100,000 – 400,000
Plastic (Polycarbonate)	50	100 – 1000	Shear in Thin Sections	5,000 – 50,000

Fundamental Formulas for Calculation of Shear Force (by Cutting)

Shear force calculation involves understanding the internal forces that resist sliding failure along a plane. The primary formula relates shear force (V), shear stress (τ), and cross-sectional area (A).

Basic Shear Force Formula:

V = τ × A

V: Shear force (Newtons, N)
τ: Shear stress (Pascals, Pa or N/m²)
A: Cross-sectional area resisting shear (m²)

Shear stress (τ) can be derived from the applied load and material properties. For uniform shear, τ is often taken as the maximum allowable shear stress of the material.

Shear Stress from Applied Force:

τ = V / A

Where the shear force V is known, and the cross-sectional area A is the area over which the force acts.

Shear Force in Beams

For beams subjected to loads, shear force varies along the length. The shear force at a section is the algebraic sum of vertical forces to one side of the section.

Shear Force at a Section:

V(x) = ΣFvertical (left or right of section x)

Where V(x) is the shear force at distance x from the beam’s support.

Shear Force in Circular Shafts (Torsion)

In shafts subjected to torsion, shear stress is related to torque (T), radius (r), and polar moment of inertia (J).

Shear Stress due to Torsion:

τ = (T × r) / J

T: Applied torque (N·m)
r: Radius at the point of interest (m)
J: Polar moment of inertia (m⁴)

Polar moment of inertia for a solid circular shaft:

J = (π × d4) / 32

d: Diameter of the shaft (m)

Shear Force in Bolted Connections

Bolted joints often experience shear force distributed over the bolt cross-section.

Shear Force per Bolt:

Vbolt = F / n

F: Total applied shear force (N)
n: Number of bolts sharing the load

Shear stress on a bolt:

τ = Vbolt / Abolt

A_bolt: Cross-sectional area of the bolt (m²)

Shear Force in Composite Sections

For composite materials or sections, shear force calculation requires summing contributions from each material, considering their shear moduli.

Equivalent Shear Force:

V = Σ (τi × Ai)

τ_i: Shear stress in material i
A_i: Cross-sectional area of material i

Detailed Explanation of Variables and Typical Values

Shear Force (V): The internal force resisting sliding failure, measured in Newtons (N). Typical values depend on load and structure size, ranging from a few hundred to millions of Newtons.
Shear Stress (τ): Stress caused by shear force, measured in Pascals (Pa). For structural steel, typical allowable shear stress is about 0.6 times the yield strength, e.g., 250 MPa for A36 steel.
Cross-Sectional Area (A): The area over which shear acts, measured in square meters (m²). For beams, this is often the web area; for bolts, the tensile stress area.
Torque (T): Twisting moment applied to shafts, measured in Newton-meters (N·m). Typical values depend on machine power and shaft size.
Radius (r): Distance from the center to the point of interest in shafts, measured in meters (m).
Polar Moment of Inertia (J): Geometric property indicating resistance to torsion, measured in m⁴. For circular shafts, calculated from diameter.
Number of Bolts (n): Count of bolts sharing the shear load. More bolts reduce shear per bolt.
Cross-Sectional Area of Bolt (A_bolt): Area resisting shear, typically the tensile stress area, measured in m².

Real-World Application Examples of Shear Force Calculation by Cutting

Example 1: Shear Force in a Simply Supported Steel Beam with Central Load

A simply supported steel beam of length 6 meters carries a central point load of 20 kN. The beam’s cross-sectional web area is 2000 mm². Calculate the maximum shear force and the shear stress in the web.

Step 1: Calculate Maximum Shear Force (V_max)

For a simply supported beam with a central point load, the maximum shear force at supports is half the load:

Vmax = P / 2 = 20,000 N / 2 = 10,000 N

Step 2: Convert Cross-Sectional Area to m²

A = 2000 mm2 = 2000 × 10-6 m2 = 0.002 m2

Step 3: Calculate Shear Stress (τ)

τ = V / A = 10,000 N / 0.002 m2 = 5,000,000 Pa = 5 MPa

Interpretation: The shear stress of 5 MPa is well below the typical allowable shear stress for structural steel (~150 MPa), indicating the beam is safe under this load.

Example 2: Shear Force in a Circular Shaft under Torsion

A solid circular steel shaft with diameter 50 mm transmits a torque of 500 N·m. Calculate the maximum shear stress in the shaft.

Step 1: Calculate Polar Moment of Inertia (J)

J = (π × d4) / 32 = (3.1416 × (0.05 m)4) / 32 = (3.1416 × 6.25 × 10-7) / 32 ≈ 6.14 × 10-8 m4

Step 2: Calculate Radius (r)

r = d / 2 = 0.05 m / 2 = 0.025 m

Step 3: Calculate Shear Stress (τ)

τ = (T × r) / J = (500 N·m × 0.025 m) / 6.14 × 10-8 m4 ≈ 2.03 × 108 Pa = 203 MPa

Interpretation: The maximum shear stress is 203 MPa, which should be compared to the material’s shear strength. For typical steel with yield strength ~250 MPa, this is near the limit, indicating the shaft is highly stressed.

Additional Considerations and Advanced Topics

Shear force calculation by cutting is not limited to simple geometries or static loads. Advanced analysis includes:

Variable Cross-Sections: Shear force varies with changing geometry; requires integration or numerical methods.
Dynamic Loading: Time-dependent loads induce fluctuating shear forces, requiring fatigue analysis.
Shear Flow in Thin-Walled Sections: Shear force distributed as shear flow (force per unit length), important in aerospace and automotive structures.
Combined Stresses: Shear combined with bending, axial, or torsional stresses requires interaction formulas for safety.
Finite Element Analysis (FEA): Numerical simulation to calculate shear forces and stresses in complex structures.

For authoritative references and design codes, consult:

ASTM International – Material standards and testing methods.
American Society of Civil Engineers (ASCE) – Structural design guidelines.
American Institute of Steel Construction (AISC) – Steel design manuals.
International Organization for Standardization (ISO) – Global standards for materials and testing.

Summary of Key Points for Shear Force Calculation by Cutting

Shear force is the internal force resisting sliding failure along a plane.
Calculation requires knowledge of applied loads, geometry, and material properties.
Formulas vary by application: beams, shafts, bolted joints, and composite sections.
Material shear strength and cross-sectional area are critical parameters.
Real-world examples demonstrate practical application and safety verification.
Advanced topics include dynamic loading, variable geometry, and numerical methods.

Mastering shear force calculation by cutting is fundamental for engineers designing safe, efficient structures and mechanical components.