If the degree of the shear force diagram is x, then the:

Explanation:

Relationship Between Shear Force, Bending Moment, and Loading Curve:

In the context of structural mechanics, the relationship between the shear force diagram (SFD), bending moment diagram (BMD), and the loading curve is governed by fundamental principles of calculus. These relationships are as follows:

• The loading curve represents the distribution of external forces (loads) acting on the beam.
• The shear force is the integral of the loading curve, indicating how the applied loads are distributed along the beam.
• The bending moment is the integral of the shear force, showing how the beam resists bending under the applied loads.

Thus, the degree of the polynomial representing each diagram follows a logical pattern:

• If the degree of the loading curve is x, the shear force diagram will have a degree of x + 1.
• If the degree of the shear force diagram is x, the bending moment diagram will have a degree of x + 1.

To summarize:

• Loading Curve: Degree = x
• Shear Force Diagram: Degree = x + 1
• Bending Moment Diagram: Degree = x + 2

1. Loading Curve:

The loading curve is represented as a polynomial function of degree x:
w(x) = a₀ + a₁x + a₂x² + ... + aₓx^x

2. Shear Force Diagram:

The shear force diagram is the integral of the loading curve:
V(x) = ∫w(x) dx = (a₀x) + (a₁x²/2) + (a₂x³/3) + ... + (aₓx^x+1/(x+1))

The degree of V(x) is x + 1.

3. Bending Moment Diagram:

The bending moment diagram is the integral of the shear force diagram:
M(x) = ∫V(x) dx = (a₀x²/2) + (a₁x³/6) + (a₂x⁴/12) + ... + (aₓx^x+2/((x+1)(x+2)))

The degree of M(x) is x + 2