# 物理代写|结构力学代写Structural Mechanics代考|CE310

## 物理代写|结构力学代写Structural Mechanics代考|Final Remarks

A good choice of redundant bars can reduce the level of calculation availed by Virtual Work. The middle lower bar in Fig. $5.2$ connects to symmetrical sub-structures on either side, which have the same statical properties from the overall loading – and only one side needs to be evaluated. Declaring the diagonal bar in Fig. $5.1$ to be redundant enabled the general set of equilibrium values to be recycled at various calculation stages.

However, the redundant bar in Fig. $5.2$ offers a more subtle point. A particular equilibrium solution arises in Table $5.4$ from $R$ taking any numerical value for the loads applied. If zero is set, the sub-structures are effectively disconnected from each other, and whatever the applied loading, their bar tensions are automatically easier to calculate. Thus, we seek out redundant members that connect simpler but isolated sub-structures.

Symmetry, as usual, complements this assessment. For example, the eight-bar truss in Fig. 5.4(a) has two redundancies, $b+r-D \cdot j=8+4 \times 2-2 \times 7=2=s$. The pair of horizontal bars are prime redundancy candidates, connected to three separate sub-structures – two being the same whatever the applied loading, Fig. 5.4(b).

But we cannot ‘extract’ any two redundant bars in this thought process; whatever partial structure is left behind should still be determinate without local mechanisms. The redundant declaration in Fig. 5.4(c) leaves the top bar ‘unsupported’, which is inadmissible.

Finally, the flip side of Virtual Work is equally important; of finding displacements or bar extensions. Indeterminate structures allow us to select the simplest virtual equilibrium set, ideally with as many zero bar tensions as possible, making $\Sigma T^* \cdot e$ easier to compute. This is the strength of the Virtual Work method, and can be verified by choosing more than one virtual equilibrium set for the same problem: the same real displacement or bar extension will emerge.

## 物理代写|结构力学代写Structural Mechanics代考|Why Does the Lower Bound Theorem Work

Rather, why should any equilibrium solution which does not violate permanent yielding of the material afford safe working of a structure? The answer is, of course, imbued by the mathematical formalism of the Lower Bound Theorem, which we now discuss informally.

Application of the Lower Bound Theorem relies upon ductility, where material yielding permits enough plastic straining for a pronounced structural manifestation. Metals are fortuitously, and mostly, ductile, but other Engineering materials, e.g. concrete, have to ‘contrive’ ductility in special cases, namely when there is insufficient metal reinforcement, which yields before catastrophic brittle failure of the concrete. Crucially, whatever the mechanism of yielding for our material, its ductility proceeds at constant stress.

A (metal) bar under tension will yield everywhere uniformly, giving a simple connection to structural ductility. Yielding in bending, however, is more complex, first occurring at the outermost fibres of the cross-section. Sustained bending causes plastic deformation to encroach from the top and bottom of the section towards the neutral axis, which strictly becomes the equal areas axis in the limit (see Chapter 15).

Determining the precise extent of plastic deformation must consider all three structural imperatives, but eventually full plasticity is achieved from being ductile. In practice though, excessive deformation tends to localise rather than to pervade. In tensile bars, we see familiar necking at a given cross-section, and in bending, most of the beam off-loads away from concentrated deformation into a so-called plastic hinge.
But therein lies the paradox: why is ductility, which implies significant yield strains, a prerequisite for applying the Lower Bound Theorem, in order to return a safe loading limit where the structure has not collapsed, i.e. not yielded critically (or entirely)?
First, we note that a safe load does not necessarily invoke a purely elastic response; part or parts of the structure can have yielded already without substantive deformation and no local collapse. As the loading rises, elastic stresses elsewhere build up even though the yielded regions have saturated in terms of stress levels – but not strain levels from being ductile. This process of redistribution, as it is known, maintains the capacity for the load increasing up to its ultimate value, just before collapse occurs.
A statically determinate structure is governed, however, by a single equilibrium solution. Its internal stresses are known exactly when material yielding begins, which defines a single limit of safe loading without any redistribution. Geometrical compatibility is automatically and separately satisfied.

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## 物理代写|结构力学代写结构力学代考|为什么下界定理有效

A(金属)棒在拉力下会均匀屈服，这与结构延展性有简单的联系。然而，弯曲屈服更复杂，首先发生在截面的最外层纤维。持续弯曲导致塑性变形从截面的顶部和底部向中性轴侵蚀，在极限处严格成为等面积轴(见第15章)

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