Structural Loads and Support Reactions in Engineering Mechanics

Uniformly Distributed Load (UDL) and Varying Load (UVL)

Converting Distributed Loads to Equivalent Point Loads

1. Uniformly Distributed Load (UDL)

Definition: A load that is evenly spread across a specific length ($L$) of a beam or structure.

Magnitude of Equivalent Point Load ($W$):

$$W = w \times L$$

• $w$: Intensity of UDL (N/m or kN/m)
• $L$: Length over which the UDL acts

The equivalent point load $W$ acts at the geometric center of the distributed load (at $L/2$ from either end).

UDL Diagram and Equivalent Point Load:

|‾‾‾‾‾‾‾‾‾‾|
| w N/m |
|___________|

Equivalent:

| | ↓ W = w × L
| | (at L/2)
|_______________

2. Uniformly Varying Load (UVL)

Definition: A load whose intensity varies linearly across the beam length, typically forming a triangular shape (zero at one end, maximum at the other).

Magnitude of Equivalent Point Load ($W$):

$$W = \frac{1}{2} \times w_{max} \times L$$

• $w_{max}$: Maximum load intensity at one end
• $L$: Length of the load

The equivalent point load $W$ acts at the centroid of the triangular load distribution, which is $L/3$ from the end where the load is maximum.

Free Body Diagrams (FBD) and Force Classification

1. Free Body Diagram (FBD)

A Free Body Diagram is a simplified graphical representation used to isolate a single object (body) and show all external forces acting upon it. The object is typically represented by a simple shape, and all forces are shown using arrows pointing in the direction they act.

2. Active Forces

These are the actual forces acting on the body that cause motion or tend to cause motion.

Examples:

• Weight ($W = mg$)
• Applied external force (push or pull)
• Wind force
• Tension in a rope (if pulling)

3. Reactive Forces (Reaction Forces)

These are forces provided by supports or surfaces that oppose the active forces, ensuring the body remains in equilibrium.

Examples:

• Normal force
• Frictional force
• Reactions at a hinge, roller, or fixed support

Two-Force and Multi-Force Members

1. Two-Force Member

Definition: A structural body or member with only two forces acting on it. These forces must act at only two points. For the member to be in equilibrium, the forces must be:

• Equal in magnitude.
• Opposite in direction.
• Collinear (acting along the same line of action).

Example: A straight link in a truss. The member is in pure tension or compression.

2. Multi-Force Member

Definition: A structural body with more than two forces acting on it, or a body subjected to forces acting at three or more points. It may also include applied moments.

Equilibrium Requirement: Requires satisfying both force equilibrium and moment equilibrium.

Example: A beam with supports and a load in the middle. Since it has three or more forces (reactions plus load), it is a multi-force member.

Structural Supports and Reaction Components

The following supports constrain the movement of a body and develop specific reaction components to maintain equilibrium.

1. Roller Support (Simple Roller)

Constraints: Allows rotation and horizontal movement. Restricts vertical movement.

Reactions Developed: 1 Reaction

• Vertical reaction force ($R_v$) only.

2. Rocker Support

Constraints: Similar to a roller support. Allows horizontal movement and some rotation.

Reactions Developed: 1 Reaction

• Vertical reaction force ($R_v$) only.

3. Hinged or Pinned Support

Constraints: Prevents movement in both vertical and horizontal directions. Allows rotation.

Reactions Developed: 2 Reactions

• Horizontal reaction force ($R_h$)
• Vertical reaction force ($R_v$)

4. Fixed Support (Cantilever)

Constraints: Restricts all movement (translation) and prevents rotation.

Reactions Developed: 3 Reactions

• Horizontal reaction force ($R_h$)
• Vertical reaction force ($R_v$)
• Moment reaction ($M$)