To determine Find the magnitudes of P 1 and P 2 . Explanation Given information: The vertical displacements of end A relative to B ( δ A / B ) is − 0.08 in . The vertical displacements of B relative to C ( δ B / C ) is − 0.1 in . The Young’s modulus of the concrete ( E ) is 4.2 × 10 3 ksi . The area of the cross section of section a-a ( A A B ) is 6 in . × 6 in . . The area of the cross section of section b-b ( A B C ) is 10 in . × 10 in . . The vertical height of segments AB and BC ( L A B and L B C ) is 10 ft . Assumption: Apply the sign convention for calculating the equations of equilibrium as given below: For the horizontal forces under equilibrium condition, take the force acting towards the right side as positive ( → + ) and the force acting towards the left side as negative ( ← − ) . For the vertical forces under equilibrium condition, take the upward force as positive ( ↑ + ) and the downward force as negative ( ↓ − ) . For moment equilibrium condition, take the clockwise moment as negative and the counter clockwise moment as positive. Calculation: Convert the unit for the length ( L A B and L B C ) : L A B = L B C = 10 ft × 12 in . 1 ft = 120 in . Draw the free body diagram of segment AB as in Figure 1. Refer to Figure 1. Here, P A B is the normal force developed in segment AB and P 1 is the force that acts at point A . Calculate the vertical forces by applying the equation of equilibrium as follows: ∑ F y = 0 P A B + P 1 + P 1 = 0 P A B + 2 P 1 = 0 P A B = − 2 P Draw the free body diagram of the segment BC as in Figure 2. Refer Figure 2. Here, P B C is the normal force developed in the segment BC and P 2 is the force that acts at point B . Calculate the vertical forces by applying the equation of equilibrium as follows: ∑ F y = 0 P B C + P 1 + P 1 + P 2 + P 2 = 0 P B C + 2 P 1 + 2 P 2 = 0 P B C = − ( 2 P 1 + 2 P 2 ) (1) Calculate the cross-sectional area of the segment AB ( A A B ) as follows: A A B = 6 in . × 6 in . = 36 in . 2 Calculate the cross-sectional area of the segment BC ( A B C ) as follows: A A B = 10 in

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Buckling of columns

The column is described as a vertical member of the structure that carries an axial compressive load, and buckling is a lateral deformation of the column under the application of load.

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Chapter 4.2, Problem 4.8P

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Find the magnitudes of P1 and P2.

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