APPM221: Shear Forces and Bending Moments

Study Note: When deriving general expressions for external and internal forces and moments, strict adherence to sign conventions is required to avoid sign compensation errors.

1. General Expressions and Sign Conventions

External Forces: \( \Sigma F \uparrow + \), \( \Sigma M \circlearrowleft + \)

Internal Forces: \( \Sigma F \downarrow + \), \( \Sigma M \circlearrowleft + \)

For a complete beam subjected to a continuous load \( w(x) \), the fundamental differential relationships are:

\frac{dM(x)}{dx} = V(x) \] \[ V(x) = \int w(x)dx + C_1 \] \[ M(x) = \int \left( \int w(x)dx \right) dx + C_1x + C_2

Methodology: Solve for the constants of integration (\( C_1 \) and \( C_2 \)) using internal sign conventions and the specific boundary conditions of the beam.

2. Dealing with Load Discontinuities

We use singularity functions like the step function \( u(x) \) and the Dirac delta function \( \delta(x) \) to model jumps, point loads, and concentrated moments in a single continuous equation.

A. Continuous Load with a Jump (at \( x = a \))

If the load \( w_1(x) \) experiences a jump of magnitude \( A \) at \( x=a \):

w(x) = w_1(x) + A \cdot u(x-a) \] \[ V(x) = \int w_1(x)dx + A(x-a)u(x-a) + C_1 \] \[ M(x) = \int \left( \int w_1(x)dx \right) + \frac{A}{2}(x-a)^2 u(x-a) + C_1x + C_2

B. Point Load \( P \) (at \( x = a \))

For an applied concentrated point load \( P \):

w(x) = w_1(x) + P \cdot \delta(x-a) \] \[ V(x) = \int w_1(x)dx + P \cdot u(x-a) + C_1 \] \[ M(x) = \int \left( \int w_1(x)dx \right) + P(x-a)u(x-a) + C_1x + C_2

Internal boundary condition: \( V(a-\epsilon) = -P + V(a+\epsilon) \)

C. Concentrated Couple / Moment \( \Lambda \) (at \( x = a \))

For an applied concentrated moment \( \Lambda \) (counter-clockwise positive):

w(x) = w_1(x) - \Lambda \cdot \delta'(x-a) \] \[ V(x) = \int w_1(x)dx - \Lambda \cdot \delta(x-a) + C_1 \] \[ M(x) = \int \left( \int w_1(x)dx \right) - \Lambda \cdot u(x-a) + C_1x + C_2

Internal boundary condition: \( M(a-\epsilon) = \Lambda + M(a+\epsilon) \)

3. The "All-in-One" Beam Example

The consolidated equations for this specific beam configuration are:

w(x) = w_1(x) + A \cdot u(x-a) + P \cdot \delta(x-b) - \Lambda \cdot \delta'(x-c) \] \[ V(x) = \int w_1(x)dx + A(x-a)u(x-a) + P \cdot u(x-b) - \Lambda \cdot \delta(x-c) + C_1 \] \[ M(x) = \int \left( \int w_1(x)dx \right) + \frac{A}{2}(x-a)^2 u(x-a) + P(x-b)u(x-b) - \Lambda \cdot u(x-c) + C_1x + C_2

4. Continuity and Differentiability Summary

By analyzing the singularity functions applied at \( x=a \), \( x=b \), and \( x=c \), we can summarize the mathematical behavior of the Load \( w(x) \), Shear \( V(x) \), and Moment \( M(x) \) diagrams.

Function	Behavior at \( x=a \) (Jump \( A \))	Behavior at \( x=b \) (Point Load \( P \))	Behavior at \( x=c \) (Couple \( \Lambda \))
Load \( w(x) \)	Jump discontinuity	Removable discontinuity (Dirac delta)	Not differentiable
Shear \( V(x) \)	Continuous	Jump discontinuity	Removable discontinuity
Moment \( M(x) \)	Differentiable (Salient point in graph)	Not differentiable (Salient point in graph)	Jump discontinuity

APPM221 Summary of Proofs