Max Bending Moment of Rigidly Clamped Beam

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SUMMARY

The maximum bending moment (Mmax) for a rigidly clamped beam with a point load (F) at its center is derived using the formula Mmax = Fx/2 - FL/8. This equation incorporates the vertical reactions at the supports, which are each equal to F/2. The term FL/8 arises from the moment distribution along the beam, specifically accounting for the fixed end conditions. Understanding this derivation is crucial for accurately analyzing bending moments in structural engineering applications.

PREREQUISITES
  • Understanding of beam mechanics and bending moment theory
  • Familiarity with rigid body equilibrium and reaction forces
  • Knowledge of moment distribution methods in structural analysis
  • Basic proficiency in calculus for maximizing functions
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This discussion is beneficial for graduate engineering students, structural engineers, and professionals involved in beam design and analysis, particularly those focusing on fixed-end conditions and bending moment calculations.

eddie135
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Dear Experts,

I am a gradute engineer posting here for the first time. I have a query regarding the simple problem to which i cannot seem to get the correct Max bending moment. Here is the problem.

A beam of length L rigidly clamped at both ends with a point load F in the centre, i.e, at location l/2
I calculate the vertical reactions to be f/2 on each end.
Now if we take a section through the beam at a distance x from the left hand side and summing moments about this section, i get the following

M(x) = F(x - L/2) + M + R1x

where R1 is the left hand vertical force which is equal to f/2 and M which is the fixing moment due to LHS of beam being rigidly clamped. From here I get lost. The Max bending moment from various books for this scenario is given to be

Mmax = Fx/2 - FL/8

I am baffled as to where the 8 comes from.

I would really appreciated it if anyone can shed some light on how this max moment is derived.

Looking forward to hearing from you.

Best Regards
Eddie
 
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How can the maximum bending moment be a function of x, if the moment is maximized with respect to x?
 

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