## Chapter 14

## Q. 14.6.3

## Q. 14.6.3

(Cambridge University Engineering Tripos: Preliminary Examination for Part I (1975).) Figs. 14.6-7(a) and (b) show respectively the dimensions of a propped uniform cantilever and its cross-section.

(a) Determine the plastic moment of resistance of the beam section if the yield stress of the material is 250 N/mm** ^{2}** in tension and in compression.

(b) If the load

*W*may be applied at any position within the span, determine the minimum value of

*W*that will cause collapse.

(c) If the load

*W*is removed just after the formation of the collapse plastic hinges, sketch the shapes of the bending moment and shear force diagrams after the removal of the load.

## Step-by-Step

## Verified Solution

(a)The neutral axis divides the cross section into two parts of equal area as in Fig. 14.6-8(a)

\qquad \begin{matrix}\text{ Therefore }: \ \text{Lever arm } \ & =(10+5)/2=7.5 \ mm \\ \text{Plastic moment } \ M_{p} & =10\times 10\times 250\times 7.5 \\ \quad & = \underline{0.188\ kN\ m } \\ \end{matrix}(b) Consider the mechanism in Fig. 14.6-8(b), where the position of the plastic hinge B is defined by the variable x.

The work equation is

or

\qquad W = \frac{1+x}{x(1-x)}\frac{M_{p}}{L}We do not know the value of *x*, but we do know that the required value of *x* is such as to make ** W** a minimum.

Therefore x = 0.414

and W=730 N

(c) In the absence of external applied load, the beam is acted on by support reactions only (see Fig. 14.6-8(c)).

Suppose the (unknown) support reactions are ** M** and

**. The only possible**

*R*shape of the bending moment diagram is a triangle (Fig. 14.6-8(d)); the only possible shape of the shear force diagram is a rectangle (Fig. 14.6-8(e)).

COMMENTS

(1) With reference to the mechanism in Fig. 14.6-8(b) it is interesting to note that (and the reader should verify this by elastic analysis) the maximum elastic moment at ** B** is 0.174

**and occurs when**

*WL**x = 0.366*

**, as shown in Fig. 14.6-9(a).**

*L*However, the reader must not be tempted into thinking that the correct mechanism is that in Fig. 14.6-9(b). As we saw in part (b) of the solution above, the hinge ** B** in the correct mechanism is at

*0.414*

**from**

*L***.**

*A*(2) With reference to Part (c) of the question, a common error is to think that

the residual moment diagram has a ‘kink’, at the plastic hinge position.

(2) With reference to Part (c) of the question, a common error is to think that

the residual moment diagram has a ‘kink’, at the plastic hinge position.