Math Problem Statement
Solution
Let's solve the problem step by step.
(i) Total Vertical Load Acting on the Beam
-
Uniformly Distributed Load (UDL):
- UDL =
- UDL length = (from the diagram: )
- Total UDL =
The UDL acts at its center, which is at the midpoint of the 6 m span. The center point is located:
-
Point Loads:
- Point load is located from A.
- Point load is located from A.
Total vertical load:
(ii) Determining Support Reactions ( and )
We will use the equilibrium equations:
- : The sum of vertical forces equals zero.
- : The sum of moments about point A equals zero.
Step 1: Taking Moments About A
To calculate , take moments about A. Clockwise moments are positive:
Moments due to all loads:
-
UDL:
- Total UDL =
- Acts at a distance of from A.
- Moment = .
-
Point Load :
- Acts at from A.
- Moment = .
-
Point Load :
- Acts at from A.
- Moment = .
-
Reaction at B ():
- acts at from A.
- Moment = .
Equilibrium equation for moments:
Step 2: Using to Find
The sum of vertical forces is zero:
Final Results:
- Total Vertical Load:
- Reactions:
Do you want a detailed explanation of any step, or help with further concepts?
Related Questions:
- How do you calculate the location of the resultant force for a UDL?
- What is the significance of moments in equilibrium problems?
- How would the reactions change if the UDL intensity increased to ?
- Can you explain how to verify equilibrium conditions for this problem?
- How are distributed loads handled differently from point loads in calculations?
Tip: When solving for reactions, always double-check the distances and ensure all forces and moments are accounted for correctly.
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Math Problem Analysis
Mathematical Concepts
Static equilibrium
Moments
Vertical forces
Uniformly distributed load (UDL)
Point loads
Formulas
Total vertical load = Sum of all loads
Moment about a point = Force × Perpendicular distance
Sum of vertical forces (ΣFy) = 0
Sum of moments about a point (ΣM) = 0
Theorems
Equilibrium of rigid bodies
Suitable Grade Level
Grades 10-12
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