Explanation Write the condition for static instability, determinacy and indeterminacy of plane frames as follows: 3 m + r < 3 j + e c statically unstable frame (1) 3 m + r = 3 j + e c statically determinate frame (2) 3 m + r > 3 j + e c statically indeterminate frame (3) 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 e c . Find the degree of static indeterminacy ( i ) using the equation; i = ( 3 m + r ) − ( 3 j + e c ) (4) 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 e c is 0. Substitute the values in Equation (2); 3 ( 3 ) + 3 = 3 ( 4 ) + 0 12 = 12 statically determinate frame Show the free-body diagram of the entire frame as in Figure 1. Refer Figure 1, Find the vertical reaction at point A by resolving the vertical component of forces. + ↑ ∑ F y = 0 A y − 30 = 0 A y = 30 k ↑ Find the horizontal reaction at point A by resolving the horizontal component of forces. + → ∑ F x = 0 − A x + 1 2 × 3 × 24 − 1.5 × 16 = 0 − A x + 36 − 24 = 0 A x = 12 k ← Find the moment at point A by taking moment at point A . + ↺ ∑ M A = 0 M A − 1 2 × 3 × 24 × 24 3 − 30 ( 10 ) + 1.5 × 16 × ( 24 − 16 2 ) = 0 M A − 288 − 300 + 384 = 0 M A = 204 k-ft Show the free-body diagram of the members and joints of the entire frame as in Figure 2. Consider point A : Resolve the vertical component of forces. + ↑ ∑ F Y = 0 30 − A Y A B = 0 A Y A B = 30 k Resolve the horizontal component of forces. + → ∑ F X = 0 − 12 + A X A B = 0 A X A B = 12 k Take moment about the point A . + ↺ ∑ M A A B = 0 204 − M A A B = 0 M A A B = 204 k-ft Consider the member AB : Resolve the vertical component of forces. + ↑ ∑ F Y = 0 A Y A B − B Y A B = 0 30 − B Y A B = 0 B Y A B = 30 k Resolve the horizontal component of forces. + → ∑ F X = 0 A X A B − 1 2 × 3 × 24 + B X A B = 0 12 − 36 + B X A B = 0 B X A B = 24 k Take moment about the point B . + ↺ ∑ M B A B = 0 − M B A B − A X A B ( 24 ) + 1 2 × 3 × 24 × 2 × 24 3 + M A A B = 0 − M B A B − 12 ( 24 ) + 576 + 204 = 0 M B A B = 492 k-ft Consider the point B : Resolve the vertical component of forces

6th Edition

ISBN: 9780357030967

Author: KASSIMALI

Publisher: CENGAGE L

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Chapter 5, Problem 69P

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Plot the shear diagram, bending moment diagram, axial force diagram, and the qualitative deflected shape of the frame.

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