Structural Analysis (10th Edition)
Structural Analysis (10th Edition)
10th Edition
ISBN: 9780134610672
Author: Russell C. Hibbeler
Publisher: PEARSON
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Question
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Chapter 15, Problem 15.1P
To determine

The reactions at supports

Expert Solution & Answer
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Answer to Problem 15.1P

The support reaction Q4 is 17.495kN.

The support reaction Q5 is 7.5kN.

The support reaction Q6 is 5kN-m.

Explanation of Solution

Given:

The flexural rigidity, EI of the beam is constant.

Concept Used:

Write the expression for fixed end moments in a beam with uniformly distributed load.

   FEM=wl212 ...... (I).

Here, the fixed end moment is FEM, the load on the beam is w and the length of the beam is l.

Write the expression for fixed end moments in a beam with point load at mid-span.

   FEM=wl8 ...... (II)

Write the expression for load matrix.

   Q=kD ...... (III)

Here, Q is the load matrix, k is the stiffness and D is the displacement.

Calculations:

Calculate the fixed end moment at 1 using Equation (I).

Substitute 5kN/m for w and 2m for l in Equation (I).

   FEM1=(5 kN/m)×( 2m)212=(20kN-m)12=1.67kN-m

Write the force matrix from the known force.

   Q=[51.6671.667Q45Q5Q6]

Write the displacement matrix for the known displacement.

   D=[D1D2D3000]

Develop the stiffness matrix.

   k=EI[ 12 L 3 6 L 2 12 L 3 6 L 2 6 L 2 4L6 L 2 2L 12 L 3 6 L 2 12 L 3 6 L 2 6 L 2 2L6 L 2 4L]

Calculate the member stiffness matrix for member 1.

   k1=EI[ 12 L 3 6 L 2 12 L 3 6 L 2 6 L 2 4L6 L 2 2L 12 L 3 6 L 2 12 L 3 6 L 2 6 L 2 2L6 L 2 4L]1243 ...... (IV)

Substitute 2m for L in Equation (IV).

   k1=EI[ 12 2 3 6 2 2 12 2 3 6 2 2 6 2 2 4 2 6 2 2 2 2 12 2 3 6 2 2 12 2 3 6 2 2 6 2 2 2 2 6 2 2 4 2 ]1243=EI[ 1.5 1.5 1.5 1.5 1.52 1.51 1.5 1.5 1.5 1.5 1.51 1.52]1243

Calculate the member stiffness matrix for member 2.

   k2=EI[ 12 L 3 6 L 2 12 L 3 6 L 2 6 L 2 4L6 L 2 2L 12 L 3 6 L 2 12 L 3 6 L 2 6 L 2 2L6 L 2 4L]4356 ...... (V)

Substitute 2m for L in Equation (V).

   k2=EI[ 12 2 3 6 2 2 12 2 3 6 2 2 6 2 2 4 2 6 2 2 2 2 12 2 3 6 2 2 12 2 3 6 2 2 6 2 2 2 2 6 2 2 4 2 ]4356=EI[ 1.5 1.5 1.5 1.5 1.52 1.51 1.5 1.5 1.5 1.5 1.51 1.52]4356

Calculate the stiffness matrix for the beam.

   K=k1+k2 ...... (VII)

Substitute the values of k1 and k2 in Equation (VII).

   k=EI[ 1.5 1.5 1.5 1.5 1.52 1.51 1.5 1.5 1.5 1.5 1.51 1.52]1243+EI[ 1.5 1.5 1.5 1.5 1.52 1.51 1.5 1.5 1.5 1.5 1.51 1.52]4356k=EI[ 1.5 1.5 1.5 1.500 1.521 1.500 1.5140 1.51 1.5 1.503 1.5 1.500 1.5 1.5 1.5 1.5001 1.5 1.52]

Substitute the values of stiffness matrix and displacement matrix in Equation (III).

   [51.6671.667Q45Q5Q6]=EI[1.51.51.51.5001.5211.5001.51401.511.51.5031.51.5001.51.51.51.50011.51.52][D1D2D3000]

Solve the above matrix to get the values of displacement.

   5=EI(1.5D1+1.5D2+1.5D3) ...... (VIII)

   1.667=EI(1.5D1+2D2+D3) ...... (IX)

   1.667=EI(1.5D1+D2+4D3) ...... (X)

Solve Equations (VIII), (IX) and (X).

   D1=19.99EID2=11.6EID3=5EI

Calculate the value of Q4.

   Q45=EI(1.5D11.5D2) ...... (XI)

Substitute 19.99EI for D1 and 11.6EI for D2 in Equation (XI).

   Q45=EI((1.5× 19.99 EI)(1.5× 11.6 EI))Q4=(12.495+5)kNQ4=17.495kN

Calculate the value of Q5.

   Q5=EI(1.5D3) ...... (XII)

Substitute 5EI for D3 in Equation (XII).

   Q5=(EI(1.5× 5 EI))kNQ5=7.5kN

Calculate the value of Q6.

   Q6=EI(D3) ....... (XIII)

Substitute 5EI for D3 in Equation (XIII).

   Q6=(EI( 5 EI))kN-mQ6=5kN-m

Conclusion:

The support reaction Q4 is 17.495kN.

The support reaction Q5 is 7.5kN.

The support reaction Q6 is 5kN-m.

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