Reaction Calculator

Simply Supported Beam Reaction Calculator

Use this calculator to determine the vertical support reactions (RA and RB) for a simply supported beam subjected to a single point load and/or a uniformly distributed load (UDL).

Calculated Reactions:

" + reactionRA.toFixed(2) + " kN

" + reactionRB.toFixed(2) + " kN

Understanding Support Reactions in Beams

In structural engineering, a beam is a structural element that primarily resists loads applied laterally to the beam's axis. These loads induce internal forces such as shear forces and bending moments within the beam. To maintain equilibrium, the supports of the beam exert forces back onto the beam, known as support reactions.

What are Support Reactions?

Support reactions are the forces and moments developed at the supports of a structure to counteract the applied external loads and maintain static equilibrium. For a simply supported beam, which rests on two supports (typically a pin support at one end and a roller support at the other), there are usually two vertical reactions to calculate: one at each support.

Types of Supports:

• Pin Support: Prevents translation in both horizontal and vertical directions but allows rotation. It provides two reaction components: one horizontal and one vertical. For simply supported beams with only vertical loads, the horizontal reaction is typically zero.
• Roller Support: Prevents translation only in the direction perpendicular to the surface it rests on (usually vertical) but allows both horizontal translation and rotation. It provides only one reaction component: a vertical force.

Our calculator focuses on the vertical reactions for a simply supported beam, which is a common and fundamental case in structural analysis.

Why are Reactions Important?

1. Equilibrium: Reactions ensure that the beam remains in static equilibrium, meaning it is neither accelerating nor rotating. The sum of all forces and moments acting on the beam must be zero.
2. Design: Knowing the support reactions is the first critical step in designing a beam. These forces determine the shear forces and bending moments throughout the beam, which are essential for selecting appropriate beam materials, cross-sections, and dimensions to prevent failure.
3. Foundation Design: The reactions are transmitted to the foundations or supporting structures, so their magnitudes are crucial for designing those elements as well.

How are Reactions Calculated? (Principles of Static Equilibrium)

The calculation of support reactions relies on the three fundamental equations of static equilibrium for a 2D system:

1. Sum of Vertical Forces = 0 (ΣFy = 0): The sum of all upward forces must equal the sum of all downward forces.
2. Sum of Horizontal Forces = 0 (ΣFx = 0): The sum of all forces acting to the left must equal the sum of all forces acting to the right. (For vertical loads only, this is usually zero).
3. Sum of Moments = 0 (ΣM = 0): The sum of all clockwise moments about any point must equal the sum of all counter-clockwise moments about the same point.

Formulas Used in This Calculator:

For a simply supported beam of span 'L' with a point load 'P' at distance 'a' from the left support (A), and a uniformly distributed load 'w' from x1 to x2:

1. Sum of Vertical Forces:
   RA + RB = P + (w * (x2 - x1))
2. Sum of Moments about Left Support (A):
   Taking moments about support A eliminates RA from the equation, allowing us to solve for RB directly. Clockwise moments are typically considered positive.
   (P * a) + (w * (x2 - x1) * ((x1 + x2) / 2)) - (RB * L) = 0
   From this, we derive:
   RB = [(P * a) + (w * (x2 - x1) * ((x1 + x2) / 2))] / L
3. Solve for RA:
   Once RB is known, RA can be found using the sum of vertical forces equation:
   RA = P + (w * (x2 - x1)) - RB

Where:

• RA: Vertical Reaction at Left Support (kN)
• RB: Vertical Reaction at Right Support (kN)
• L: Beam Span (m)
• P: Point Load (kN)
• a: Distance of Point Load from Left Support (m)
• w: Uniformly Distributed Load magnitude (kN/m)
• x1: Start position of UDL from Left Support (m)
• x2: End position of UDL from Left Support (m)
• (x1 + x2) / 2: Centroid of the UDL (where its equivalent point load acts)

Example Calculation:

Let's use the default values in the calculator:

• Beam Span (L) = 10 m
• Point Load (P) = 20 kN
• Distance of Point Load (a) = 3 m
• UDL Magnitude (w) = 5 kN/m
• UDL Start (x1) = 2 m
• UDL End (x2) = 7 m

Step 1: Calculate total equivalent UDL force and its centroid.

• UDL Length = x2 – x1 = 7 m – 2 m = 5 m
• Equivalent UDL Point Load (P_UDL) = w * UDL Length = 5 kN/m * 5 m = 25 kN
• Centroid of UDL from Left Support = x1 + (UDL Length / 2) = 2 m + (5 m / 2) = 2 m + 2.5 m = 4.5 m

Step 2: Sum moments about Left Support (A) to find RB.

• Moment from Point Load = P * a = 20 kN * 3 m = 60 kNm (clockwise)
• Moment from UDL = P_UDL * Centroid = 25 kN * 4.5 m = 112.5 kNm (clockwise)
• Moment from RB = RB * L = RB * 10 m (counter-clockwise)

ΣM_A = 0:

60 kNm + 112.5 kNm – (RB * 10 m) = 0

172.5 kNm = RB * 10 m

RB = 172.5 kNm / 10 m = 17.25 kN

Step 3: Sum vertical forces to find RA.

• Total Downward Force = P + P_UDL = 20 kN + 25 kN = 45 kN

ΣFy = 0:

RA + RB = Total Downward Force

RA + 17.25 kN = 45 kN

RA = 45 kN – 17.25 kN = 27.75 kN

Thus, for the given inputs, the reactions are RA = 27.75 kN and RB = 17.25 kN.