Point Loads on Beams

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1. Introduction to Structural Members

Beams are fundamental structural elements designed to resist loads applied perpendicular to their longitudinal axis. The study of how point loads affect beams is critical in understanding responses such as support reactions, internal shear forces, and bending moments.

1.1 Definition: Point Load

A point load (or concentrated load) is a force applied at a single, specific location along the beam. In practice, true point loads are idealizations, as real loads act over finite areas; nonetheless, the point load model is useful for analysis of structural elements.

2. Types of Beams and Supports

2.1 Common Beam Types

  1. Simply Supported Beam: Supported at both ends, allowing rotation but not translation.
  2. Cantilever Beam: Fixed at one end, free at the other.
  3. Overhanging Beam: Extends beyond its supports at one or both ends.

2.2 Support Types

  • Pinned Support: Allows rotation, restricts translation.
  • Roller Support: Allows rotation and horizontal translation, restricts vertical movement.
  • Fixed Support: Restricts translation and rotation.

3. Analysis of Point Loads on Beams

3.1 Reactions at Supports

Goal: To determine the reactions at the supports due to a given point load.

Example Problem 1: Simply Supported Beam with a Single Point Load

Consider a beam of length L, simply supported at both ends, with a point load P applied at a distance a from the left support.

Step 1: Free Body Diagram (FBD)

  • Draw the beam, mark support points A (left) and B (right).
  • Apply point load P at distance a from A, L – a from B.

Step 2: Equilibrium Equations

  • Sum of vertical forces:

\sum F_y = 0: R_A + R_B - P = 0

Sum of moments about A:

\sum M_A = 0: R_B \cdot L - P \cdot a = 0

Step 3: Solve for Reactions

  • From moments:

R_B = \frac{P \cdot a}{L}

Substitute in vertical force equation:

R_A = P - R_B = \frac{P (L - a)}{L}

Table: Support Reactions

SupportReaction Formula
LeftR_A = \frac{P(L-a)}{L}
RightR_B = \frac{P a}{L}

4. Shear Force and Bending Moment

4.1 Shear Force Diagram (SFD)

A shear force at a section is the sum of vertical forces to the left or right of the section. It experiences a sudden change (jump) at the location of the point load.

  • Left of point load: Shear force is R_A.
  • Right of point load: Shear force is R_A - P = -R_B.

SFD Construction Steps:

  1. Start from left end at R_A.
  2. Drop by P at the load location.
  3. Continue at -R_B to the right end.

4.2 Bending Moment Diagram (BMD)

The bending moment at a section is the sum of moments about that section due to forces on one side.

  • For location x<ax < ax<a:

M(x) = R_A \cdot x

For location x>ax > ax>a:

M(x) = R_A \cdot x - P(x-a)

Maximum Bending Moment under the point load (x=ax = ax=a):

M_{max} = R_A a = \frac{P (L-a) a}{L}

5. Worked Example

Example Problem

Given: Simply supported beam, L=6 m, point load P=10 kN at a=2 m from the left support.

Solution:

  • R_A = \frac{10 \times (6-2)}{6} = 6.67, \text{kN}
  • R_B = \frac{10 \times 2}{6} = 3.33, \text{kN}
  • Max bending moment:

M_{max} = 6.67 \times 2 = 13.33 , \text{kN}\cdot\text{m}

6. Summary Table: Key Formulas for Point Loads on Beams

QuantityFormula
Reaction at left supportR_A = \frac{P(L-a)}{L}
Reaction at right supportR_B = \frac{P a}{L}
Shear just left of loadR_A
Shear just right of load-R_B
Max bending moment (under P)M_{max} = \frac{P(L-a)a}{L}
Bending moment at section xM(x) = R_A x (for x < a)
M(x) = R_A x - P(x-a) (for x > a)

7. Interpretation and Physical Significance

  • At the application point of a load, shear force exhibits a discontinuity (jump).
  • Bending moment profile is piecewise linear, with the maximum under the point load.
  • These insights are essential for structural design and safety assessments.

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