Question

problem 1
determine the normal force, shear force, and moment at
point c. take that ( p _ { 1 } = 10 mathrm { kn } ) and ( p _ { 2 } = 16 mathrm { kn } ) (figure 1)
figure
the figure shows a horizontal beam with its left and right ends supported by a pin at point a and a roller at point b, respectively.
point c is in the middle of the beam ab. one half of ac has a length of 1.5 meters. a force ( p _ { 1 } ) acts downward in the middle of
ac and a force ( p _ { 2 } ) acts downward in the middle of cb.
part a
determine the normal force at point c.
express your answer to three significant figures and include the appropriate units.
part b
determine the shear force at point c
express your answer to three significant figures and include the appropriate units.

Explanation:

Step1: Calculate the reactions at supports

Take moments about point \(A\). The total length of the beam \(AB = 6m\). Let the reaction at \(B\) be \(R_{B}\).
\(\sum M_{A}=0\): \(P_{1}(1.5)+P_{2}(4.5)-R_{B}(6) = 0\)
Substitute \(P_{1}=10kN\) and \(P_{2}=16kN\)
\(10\times1.5 + 16\times4.5-6R_{B}=0\)
\(15+72 - 6R_{B}=0\)
\(87=6R_{B}\)
\(R_{B}=14.5kN\)
By vertical force equilibrium \(\sum F_{y}=0\): \(R_{A}+R_{B}-P_{1}-P_{2}=0\)
\(R_{A}+14.5 - 10 - 16=0\)
\(R_{A}=11.5kN\)

Step2: Analyze forces at point \(C\) (for normal force \(N_{C}\))

Cut the beam at point \(C\). Assume the normal force \(N_{C}\), shear force \(V_{C}\) and moment \(M_{C}\) at point \(C\).
For normal force: Since there is no axial load in the beam (all forces are vertical), \(N_{C}=0\)

Step3: Analyze shear force at point \(C\) (for \(V_{C}\))

Sum of vertical forces to the left of \(C\): \(V_{C}=R_{A}-P_{1}\)
\(V_{C}=11.5 - 10=1.5kN\)

Answer:

• Normal force \(N_{C}=0kN\)
• Shear force \(V_{C}=1.5kN\)