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How do you find the general equations for shear force and bending moment?

For example, in this question, UDL = $w = 5.4knm$ and $L = 8.6$:

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How do you find the constants of unknown from integration?

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2 Answers 2

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To find the constants, simply plug the known conditions into the equations:

For shear, V = 0 @ x = L/2

For moment, M = 0 for x = 0 and x = L

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  • $\begingroup$ How did you find out the known conditions? Why is V = 0 @ x = L/2 and why is the M = 0 for x = 0 and x = L? Are these two conditions always true in every scenario? $\endgroup$
    – CountDOOKU
    Jul 30, 2022 at 14:35
  • $\begingroup$ @CountDOOKU its because of how your constraints have been laid out. Triangle means that its possible to rotate hence M = 0 since nothing is forcing it in place. And so on... $\endgroup$
    – joojaa
    Jul 30, 2022 at 15:49
  • $\begingroup$ The conditions are true for a simply supported beam with the uniformly distributed load only. $\endgroup$
    – r13
    Jul 30, 2022 at 15:58
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Just use definite integral on the portion of the beam of interest. Usually you integrate from some point where you know the value to another, where the value is unknown. For example moment is zero at simple support and then changes as you go further from it, where the integral represents the change along the integrated beam portion.

Shear force $T(x)$ ($T_0 = \frac{w\cdot L}{2}$ supports half of the beam load): $$T(x) = T_0 - \int\limits_{0}^{x} w(x) dx = \frac{w\cdot L}{2} - w\cdot x$$

Bending moment $M(x)$ ($M_0 = 0$ due to simple support): $$M(x) = M_0 - \int\limits_{0}^{x} T(x) dx = 0-\frac{w\cdot L}{2}\cdot x + \frac{w}{2}\cdot x^2$$

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