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The position of the support point where the magnitude of the maximum moment is minimized when a distributed load applies on the beams

작성자 Uploader : milinae 작성일 Upload Date: 2023-05-23변경일 Update Date: 2023-07-05조회수 View : 2414

When the length of the beam is L (m), the length between the end of the beam and the support point is L1 (m), and the magnitude of the applied distributed load is w (kN/m), the length L1 (m) that makes the maximum moment to minimize can be found.

Reaction force at support : RA = RB = w*L/2

The magnitude of the moment increases from the end of the beam to the support point A, decreases as it passes through the support point, and increases again as it passes through the center of the beam.
Since it has an extremal value at the support point and the center of the beam, the maximum moment becomes the minimum when the magnitude of the moment at the support point and at the center are the same.

Moment at support

MA = w*L1*(L1/2)

Moment at beam center

MC = RA*(L/2-L1) - w*(L/2)*(L/4)
   = (w*L/2)*(L/2-L1) - (w/8)*L^2
   = (w/4)*(L^2-2L*L1 - (1/2)*L^2)
   = (w/4)*((1/2)L^2 - 2L*L1)

|MA| = |MC|

w*L1*(L1/2) = (w/4)*((1/2)L^2-2L*L1)

L1^2 = (1/4)L^2 - L*L1

L1^2 + L*L1 - (1/4)L^2

Using the quadratic formula, since 0 < L1, L1 is as follow.

L1 = (1/2)*(-L+sqr(2*L^2)) = (L/2)*(sqr(2)-1)


*** 참고문헌[References] ***
L1 = (L/2)*(sqr(2)-1)
변수명 Variable 변수값 Value 변 수 설 명 Description of the variable


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