5 14 Determine The Reactions At The Supports

Let's delve into the process of determining support reactions for a statically determinate structure, specifically focusing on the application of statics principles. Understanding these reactions is fundamental to structural analysis and design, ensuring structures can withstand applied loads without failure.

Statics: The Foundation of Support Reaction Calculations

Before we jump into the specifics, let's establish a solid understanding of statics. Statics is the branch of mechanics that deals with bodies at rest or moving with constant velocity. In structural engineering, it forms the basis for analyzing forces and moments acting on structures and ensuring they are in equilibrium.

Equilibrium: A structure is in equilibrium when the sum of all forces and moments acting on it equals zero. This is crucial for stability. Mathematically, equilibrium is expressed as:

• ΣFₓ = 0 (Sum of forces in the x-direction is zero)
• ΣFᵧ = 0 (Sum of forces in the y-direction is zero)
• ΣM = 0 (Sum of moments about any point is zero)

These three equations are the cornerstone of solving for unknown support reactions in two-dimensional (2D) statically determinate structures.

Statically Determinate Structures: A statically determinate structure is one where all support reactions can be determined using only the equations of static equilibrium. These structures have a balance between the number of unknowns (support reactions) and the number of independent equilibrium equations available. Common examples include simply supported beams, cantilevers, and beams with hinges.

Types of Supports and Their Reactions

Different types of supports offer different constraints and, consequently, different reaction forces. It's crucial to identify the support types accurately to correctly determine the unknowns. Here's a breakdown of common support types:

1. Roller Support: A roller support allows rotation and translation along one direction. It provides resistance only in one direction, perpendicular to the surface it rolls on. Therefore, it has one unknown reaction force, acting perpendicular to the supporting surface.

2. Hinge (or Pin) Support: A hinge support allows rotation but prevents translation in any direction. It provides resistance in both horizontal and vertical directions. Therefore, it has two unknown reaction forces: one horizontal and one vertical.

3. Fixed Support: A fixed support prevents both rotation and translation. It provides resistance in both horizontal and vertical directions and also resists rotation by providing a moment. Therefore, it has three unknown reactions: one horizontal force, one vertical force, and one moment.

The Step-by-Step Process to Determine Support Reactions

Now, let's outline the step-by-step process to determine the reactions at the supports of a statically determinate structure:

Step 1: Draw a Free Body Diagram (FBD)

The FBD is the most critical step. It's a simplified representation of the structure, showing all external forces and moments acting on it. This includes:

• Applied Loads: All external forces acting on the structure (e.g., point loads, distributed loads). Clearly indicate their magnitude and direction.
• Support Reactions: Represent each support with its corresponding reaction forces. If the direction of a reaction is unknown, assume a direction (positive x and y are common). The calculations will reveal the correct direction (a negative result simply means the assumed direction was incorrect).
• Dimensions: Clearly indicate all relevant dimensions and distances on the diagram. These are essential for calculating moments.

Example:

Let's consider a simply supported beam with a point load 'P' at its center. The beam is supported by a hinge at point A and a roller at point B.

• Applied Load: Point load 'P' acting downwards at the center of the beam.
• Support Reactions:
  • Hinge at A: Vertical reaction force Ay (assumed upwards) and horizontal reaction force Ax (assumed to the right).
  • Roller at B: Vertical reaction force By (assumed upwards).
• Dimensions: Total length of the beam = L. Distance from A to P = L/2. Distance from B to P = L/2.

Step 2: Apply the Equations of Equilibrium

Apply the three equations of static equilibrium (ΣFₓ = 0, ΣFᵧ = 0, ΣM = 0) to the FBD. This will result in a system of equations that can be solved for the unknown support reactions. The key is to choose the moment equation strategically.

• ΣFₓ = 0: Sum of all horizontal forces equals zero. This equation will help solve for horizontal reactions.
• ΣFᵧ = 0: Sum of all vertical forces equals zero. This equation will help solve for vertical reactions.
• ΣM = 0: Sum of all moments about any point equals zero. This is where strategic thinking comes in. Choose a point where several unknown forces intersect. This eliminates those forces from the moment equation, simplifying the calculations. Typically, choosing a support location as the point about which to sum moments is advantageous.

Step 3: Solve the Equations

Solve the system of equations obtained in Step 2 to determine the unknown support reactions. This may involve simple algebra or solving a system of simultaneous equations.

Step 4: Verify Your Results

Once you've calculated the support reactions, it's crucial to verify your results to ensure accuracy. You can do this by:

• Substituting the calculated reaction values back into the equilibrium equations. All equations should hold true.
• Taking the sum of moments about a different point than the one used initially. This provides an independent check of your calculations. If the sum of moments about this new point also equals zero, your results are likely correct.

Detailed Example: Simply Supported Beam with a Point Load

Let's go back to our simply supported beam example and work through the calculations.

1. Free Body Diagram: (As described above)

2. Equations of Equilibrium:

• ΣFₓ = 0: Ax = 0 (This tells us that the horizontal reaction at A is zero since there are no other horizontal forces acting on the beam.)
• ΣFᵧ = 0: Ay + By - P = 0
• ΣMᴀ = 0: (Taking moments about point A) By * L - P * (L/2) = 0

3. Solve the Equations:

• From ΣFₓ = 0, we already know Ax = 0.
• From ΣMᴀ = 0, we can solve for By:
  • By * L = P * (L/2)
  • By = (P * L) / (2 * L)
  • By = P/2
• Substitute By = P/2 into ΣFᵧ = 0:
  • Ay + (P/2) - P = 0
  • Ay = P - (P/2)
  • Ay = P/2

4. Verify Results:

• ΣFₓ = 0: 0 = 0 (Checks out)
• ΣFᵧ = 0: (P/2) + (P/2) - P = 0 => P - P = 0 (Checks out)
• ΣMʙ = 0: (Taking moments about point B) -Ay * L + P * (L/2) = 0 => -(P/2) * L + P * (L/2) = 0 => 0 = 0 (Checks out)

Therefore, the support reactions are:

• Ax = 0
• Ay = P/2 (upwards)
• By = P/2 (upwards)

This result makes intuitive sense. The vertical reactions at both supports are equal and each supports half of the applied load.

Dealing with Distributed Loads

Distributed loads are loads that are spread over a certain length of the structure. To determine support reactions, you must first convert the distributed load into an equivalent point load.

Uniformly Distributed Load (UDL): A UDL has a constant load intensity (w) over a length (L). The equivalent point load is:

• Magnitude: W = w * L
• Location: Acts at the centroid of the distributed load, which is usually at the midpoint of the loaded length for a rectangular UDL.

Triangularly Distributed Load: A triangularly distributed load has a load intensity that varies linearly. The equivalent point load is:

• Magnitude: W = (1/2) * base * height = (1/2) * L * w (where w is the maximum load intensity)
• Location: Acts at the centroid of the triangle, which is located at L/3 from the end with the maximum load intensity.

Example:

Consider a simply supported beam of length L with a uniformly distributed load 'w' over its entire length.

1. Convert UDL to a Point Load: The equivalent point load W = w * L, acting at L/2 from either support.

2. Free Body Diagram: Draw the FBD with Ay, By (vertical reactions at supports), and the equivalent point load W at the center.

3. Equations of Equilibrium:

   • ΣFₓ = 0 (No horizontal forces, so no horizontal reaction)
   • ΣFᵧ = 0: Ay + By - W = 0
   • ΣMᴀ = 0: By * L - W * (L/2) = 0

4. Solve the Equations:

   • From ΣMᴀ = 0: By = (W * L) / (2 * L) = W/2 = (w * L) / 2
   • From ΣFᵧ = 0: Ay = W - By = w * L - (w * L) / 2 = (w * L) / 2

Therefore, Ay = By = (w * L) / 2. Each support carries half of the total distributed load.

Cantilever Beams

Cantilever beams are fixed at one end and free at the other. The fixed end provides all three reactions: a vertical force, a horizontal force, and a moment.

Example:

Consider a cantilever beam of length L with a point load P at the free end.

1. Free Body Diagram: Draw the FBD with the fixed support showing reactions Ax, Ay, and MA (moment reaction). The point load P acts downwards at the free end.

2. Equations of Equilibrium:

   • ΣFₓ = 0: Ax = 0
   • ΣFᵧ = 0: Ay - P = 0
   • ΣMᴀ = 0: MA - P * L = 0 (Taking moments about the fixed end A)

3. Solve the Equations:

   • Ax = 0
   • Ay = P
   • MA = P * L

The fixed support provides a vertical reaction equal to the applied load and a moment reaction equal to the load multiplied by the length of the beam. The direction of the moment is such that it resists the rotation caused by the point load.

Common Mistakes to Avoid

• Incorrect Free Body Diagram: This is the most common source of errors. Make sure to include all forces and moments, and represent the support reactions correctly. Pay close attention to the directions of forces.
• Sign Conventions: Be consistent with your sign conventions (e.g., upward forces are positive, clockwise moments are positive). Changing sign conventions mid-calculation will lead to errors.
• Units: Ensure all units are consistent. If lengths are in meters, forces should be in Newtons (or kilonewtons), and moments in Newton-meters (or kilonewton-meters).
• Incorrectly Calculating Moments: Double-check that you are using the correct perpendicular distance when calculating moments. The moment is the force multiplied by the perpendicular distance from the force's line of action to the point about which you are taking the moment.
• Forgetting to Verify: Always verify your results to catch any errors.

Advanced Considerations

While this article focuses on statically determinate structures in 2D, it's important to acknowledge more complex scenarios:

• Statically Indeterminate Structures: These structures have more unknown reactions than available equilibrium equations. Solving them requires additional methods like the slope-deflection method, moment distribution method, or finite element analysis.
• 3D Structures: Analyzing structures in three dimensions requires considering forces and moments in all three directions (x, y, z). This involves more complex equilibrium equations and can be computationally intensive.
• Dynamic Loads: Structures subjected to dynamic loads (e.g., impact loads, vibrations) require dynamic analysis techniques, which go beyond the scope of static equilibrium.

Conclusion

Determining support reactions is a fundamental skill in structural analysis. By understanding the principles of statics, carefully drawing free body diagrams, applying the equations of equilibrium, and verifying your results, you can accurately determine these reactions for statically determinate structures. This knowledge forms the basis for more advanced structural analysis and design. Remember to practice regularly and pay attention to detail to avoid common mistakes. A solid grasp of these principles is essential for any aspiring structural engineer.