Concept explainers
Plot the shear diagram, bending moment diagram, axial force diagram, and the qualitative deflected shape of the frame.
Explanation of Solution
Write the condition for static instability, determinacy and indeterminacy of plane frames as follows:
Here, number of members is m, number of external reactions is r, the number of joints is j, and the number of elastic hinges is
Find the degree of static indeterminacy (i) using the equation;
Refer to the Figure in the question;
The number of members (m) is 3.
The number of external reactions (r) is 3.
The number of joints (j) is 4.
The number of elastic hinges
Substitute the values in Equation (2);
Show the equivalence of uniformly distributed load as in Figure 1.
Show the free-body diagram of the entire frame as in Figure 2.
Find the horizontal reaction at point A by resolving the horizontal component of forces.
Find the vertical reaction at point D by taking moment about point A.
Find the vertical reaction at point A by resolving the vertical component of forces.
Show the free-body diagram of the members and joints of the entire frame as in Figure 3.
Consider point A:
Resolve the vertical component of forces.
Resolve the horizontal component of forces.
Consider the member AB:
Resolve the vertical component of forces.
Take moment about the point B.
Consider the point D:
Resolve the vertical component of forces.
Consider the member CD:
Resolve the vertical component of forces.
Take moment about the point C.
Resolve the horizontal component of forces.
Consider the point C:
Resolve the vertical component of forces.
Resolve the horizontal component of forces.
Take moment about the point C.
Consider the section BC:
Resolve the vertical component of forces.
Resolve the horizontal component of forces.
Take moment about point B:
Plot the moment end forces of the frame as in Figure 4.
Refer to the moment end force diagram plot the shear diagram, bending moment diagram, and the axial force diagrams.
Plot the shear force diagram as in Figure 5.
The maximum bending moment occurs where the shear force changes sign.
Consider the section BEC, use the similar triangle concept.
Plot the bending moment diagram as in Figure 6.
Plot the axial force diagram as in Figure 7.
Plot the qualitative deflected shape as in Figure 8.
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Structural Analysis