# 物理代写|结构力学代写Structural Mechanics代考|ENG222

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

Ductility provides a literal safety margin for our ‘incorrect’ trial solutions compared to true behaviour, and coupled to the Lower Bound Theorem returning a smaller, safe ultimate load every time, the freedom to select equilibrium solutions is key to its efficacy.

If we are designing the truss for a fixed load in practice, a lower ultimate prediction means using bars with larger cross-sectional areas than is necessary. The truss, we say, will be over-designed in terms of being able to accommodate a higher fixed load (which hopefully is not applied in practice).

This final statement speaks to the remarkable prescience of the Lower Bound Theorem in general as a design tool. Where else, in Engineering, does an incorrect solution result in a margin of safety and increasingly so for a greater departure from the exact solution? The detriment is, of course, a more massive structure.

We will invariably over-design more complex structures, which have many more equilibrium combinations. Rather than trying to seek one of the better ones, we should aim for simplicity if pressed for time where a ‘back-of-the-envelope’ calculation is not anachronistic.

Again, we note that only the final equilibrium state matters; compatibility does not have to be assessed because of ductility, which is simply contrived. In practice, yielding leads to higher stresses after initial yielding, which would result in the junctions between phases in Fig. $6.2$ becoming more rounded, for example.

We also have a perfect structure in which bars neatly connect together without mis-fitting. There will inevitably be small misalignments with some bars having to be stretched or compressed in order to complete building the truss, giving non-zero tensions for zero loading, as we saw in Chapter $5 .$

Any state of self-stress increases the proximity to yielding but does affect the final capacity of individual members. In this regard, deliberate pre-stressing and unforeseen movements of supports dó noot subtract from the ultimatee load.

## 物理代写|结构力学代写Structural Mechanics代考|Bending Moment (and Shear Force) Diagrams

Often the scourge of undergraduate exercises, drawing bending moment profiles for slender structures is an essential skill. Their variation tells us about how a beam or column is loaded transversely as well as indicating salient values at pertinent locations where damage may be incurred from extreme loading.

In the case of, say, a steel beam, it will submit to permanent yielding of the material at these points, which concentrates into plastic hinges; collapse of the entire structure may follow depending on how the structure adjusts to more loading. Equally important are shear force diagrams – especially for concrete structures, where failure in shear often dominates that of bending.

The usual way to teach construction of bending moment (and shear force) diagrams is through piece-wise equilibrium. We select a given free body, resolve forces and take moments to establish its variation before doing the same with a different free body. We then assemble the complete profile reliably but inefficiently. We shall adopt a more holistic approach.

For small variations in bending moment, $M$, and shear force, $S$, across an element of straight beam of length $\delta x$, Fig. 7.1(a), equilibrium gives their well-known relationships to some externally applied, transverse loading intensity, $w$, as:
$$\frac{\mathrm{d} S}{\mathrm{~d} x}=w \quad \text { (a), } \frac{\mathrm{d} M}{\mathrm{~d} x}=S \quad \text { (b) } \rightarrow \quad \frac{\mathrm{d}^2 M}{\mathrm{~d} x^2}=w \quad \text { (c). }$$
These statements convey much information.
First, the directions of $M$ and $S$ obey an explicit sign convention, which we must always declare a priori. Here, $M$ is positive if the beam is locally hogging and curving upwards, and shear forces point downwards on the left side and vice versa rightside. The latter is also linked to the direction of increasing $x$, for if $x$ is reversed in measurement along the beam, so do the positive shear force directions, in order to preserve the statements above. Indeed, our choice in absolute directions for all leads to an absence of minus signs when $w$ acts downwards in the sense of gravity. We can, of course, choose a different scheme, such as one with positive sagging bending moments, which will introduce minus signs.

We can find the shear force profile by differentiating that of bending moment, if it is constructed first. If not, we may integrate shear to find bending moments, which is tantamount to finding the area underneath the shear force profile.

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## 物理代写|结构力学代写结构力学代考|弯矩(和剪力)图

$$\frac{\mathrm{d} S}{\mathrm{~d} x}=w \quad \text { (a), } \frac{\mathrm{d} M}{\mathrm{~d} x}=S \quad \text { (b) } \rightarrow \quad \frac{\mathrm{d}^2 M}{\mathrm{~d} x^2}=w \quad \text { (c). }$$

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