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In each case, calculate the reaction at A and then draw the free-body diagram o...

May 20, 2024

In each case, calculate the reaction at A and then draw the free-body diagram of segment AB of the beam in order to determine the internal loading at B.

Solution

Calculate the reaction at point A by considering the equilibrium of the entire beam. The distributed load on segment CD is

200 \, \text{N/m}

over a length of

4 \, \text{m}

, resulting in a total load of

800 \, \text{N}

. This load acts at the midpoint of segment CD, which is

2 \, \text{m}

from point C

Sum of vertical forces:

\sum F_y = 0 \Rightarrow R_A + R_C - 800 \, \text{N} = 0

where

R_A

is the reaction at A and

R_C

is the reaction at C

Sum of moments about point A:

\sum \tau_A = 0 \Rightarrow -800 \, \text{N} \times 6 \, \text{m} + R_C \times 4 \, \text{m} = 0

Solve for

R_C

R_C = \frac{800 \, \text{N} \times 6 \, \text{m}}{4 \, \text{m}} = 1200 \, \text{N}

Substitute

R_C

back into the vertical force equilibrium equation:

R_A + 1200 \, \text{N} - 800 \, \text{N} = 0 \Rightarrow R_A = -400 \, \text{N}

Draw the free-body diagram of segment AB. The internal loading at B includes the reaction force at A and the internal shear force and moment at B

Answer

The reaction at point A is

-400 \, \text{N}

, indicating a downward force.

Key Concept

Static Equilibrium: The sum of forces and moments acting on a system in equilibrium must be zero. Equations:

\sum F = 0

and

\sum \tau = 0

Explanation

By applying the principles of static equilibrium, we calculated the reaction forces at the supports and determined the internal loading at a specific point on the beam.