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  3. 구조공학 Structural

Cantilever beam, moment calculation of arbitrary point, trapezoidal distribution load

작성자 Uploader : metalheart 작성일 Upload Date: 2019-10-29변경일 Update Date: 2019-12-29조회수 View : 2960

When the trapezoidal distribution load acts on the cantilever beam as shown in the figure, the shear force BMx at any point x can be obtained as below.

Find the moment from the load acting on the right side of point x.

If P1 ≤ P2,

The linear equation of the load from point to x is

y = (P2-P1) * x / L + P1

The load can be divided into a rectangular distributed load with a height of y and a triangular distributed load with a height of P2-y.

BMx = y * (L-x) * (L-x) / 2 + ((P2-y) * (L-x) / 2) (2 (L-x) / 3)

    = (L-x) ^ 2 * (y / 2 + (P2-y) / 3)

    = (L-x) ^ 2 * (1/6) (3y + 2P2-2y)

    = (1/6) * (L-x) ^ 2 * ((P2-P1) * x / L + P1 + 2P2)

If P1 ≥ P2,

y =-(P1-P2) * x / L + P1 = (P2-P1) * x / L + P1

The load can be divided into a rectangular distributed load with a height of P2 and a triangular distributed load with a height of y-P2.

BMx = P2 * (L-x) * (L-x) / 2 + ((y-P2) * (L-x) / 2) (L-x) / 3

    = (L-x) ^ 2 * (P2 / 2 + ((P2-P1) * x / L + P1-P2) / 6)

    = (1/6) * (L-x) ^ 2 * ((P2-P1) * x / L + P1 + 2P2))

The same equation can be applied to both cases.

*** 참고문헌[References] ***
BMx = (1/6)*(L-x)^2*(P1 + 2*P2 + (P2-P1)*x/L))
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변수명 Variable 변수값 Value 변 수 설 명 Description of the variable


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