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I know why the rope sags but I don't understand why the same argument can't be made for the beam. What is it that allows the beam to stay straight when a mass is placed on it?

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    $\begingroup$ Do you know the definition of a rigid body? Or are you asking how we can approximate things as rigid bodies? $\endgroup$ Nov 5 '19 at 0:42
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The beam also sags. It just sags a lot less because it has so much more stiffness, as measured by its Young’s modulus.

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  • $\begingroup$ A string has Young's modulus also. But a string does not have resistance to bending as its second moment of area is negligible and thus the section modulus $E I \rightarrow 0$. $\endgroup$ Nov 5 '19 at 6:39
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What is it that allows the beam to stay straight when a mass is placed on it?

The beam does not stay straight.

You need to differentiate between how the beam is considered in determining static equilibrium vs how it considered in terms of deformation due to loading.

For the purpose of determining static equilibrium, the beam is considered a rigid body. It is assumed to stay straight when a mass (load) is placed on it. That allows you to determine reactions at the supports, and then vertical shear and bending moments on the beam.

But once the static equilibrium analysis is complete, you then need to determine shear and bending stresses in the beam and the associated deformations. At that point the beam is no longer considered a rigid body. It will deform depending on the properties of the beam material, which, as @G. Smith pointed out, is the materials modulus of elasticity (Young's modulus), as well as the moment of inertia of the beams cross section.

Hope this helps.

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A rope has zero negligible resistance to bending, and a beam has significant resistance to bending due to its section properties. The fundamental difference is that a rope cannot sustain any compression, while a beam can. Compression is needed for bending as one side is in tension and the other in compression.

But given a load in the middle of two supports, both the rope and beam will deform. The rope will take the shape of a triangle, whilst the beam will acquire the shape of a cubic function.

The resistance in bending is typically expressed as

$$ M = E I \tfrac{{\rm d}^2}{{\rm d}x^2} w_y(x) $$

where $M$ is the internal reaction moment (torque), $w_y(x)$ the shape of the deflection, $E$ is the modulus of elasticity and it is a material property, and finally, $I$ is the area moment (second moment of area) and it is a property of the cross-section.

Both rope and beam have the same equation for axial stretching

$$ F = E A \tfrac{{\rm d}}{{\rm d}x} w_x(x) $$

where $F$ is an axial force, $w_x(x)$ is the axial deflection, and $A$ is the cross sectional area.

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