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## Q. 14.6.1

The propped uniform cantilever in Fig. 14.6-4(a) is of plastic moment $M_{P}$. Determine the value of P at collapse using (a) the work method (b) the statical method.

## Verified Solution

(a) The work method (or virtual work method)

There are three possible collapse mechanisms, as shown in Figs. 14.6-4(b), (c) and (d). We shall calculate the collapse load for each in turn.

The work equation is

$\qquad P\left(\phi L\right)+2P(3\phi L/2)=M_{p}\phi +M_{p}(4\phi )$

Therefore                          P = 5Mp/4L

$\qquad P\left(\phi L\right)+2P(\phi L/2)=M_{p}\phi +M_{p}(2\phi )$

Therefore                            P = 3Mp/2L

$\qquad 2P(\phi L/2)=M_{p}\phi +M_{p}(2\phi )$

Therefore                             P = 3Mp/L

We have now examined all the three possible mechanisms and found that the one in Fig. 14.6-4(b) gives the lowest collapse load. This means that as the magnitude of P is gradually increased from zero, the collapse mechanism in Fig. 14.6-4(b) will be the first to form, when P reaches $5M_{p}/4L$. The other two mechanisms cannot form unless this one is prevented from forming, for example by strengthening the cross sections at points where hinges would have formed. We therefore conclude that $P = 5M_{p}/4L$ is the correct collapse value. (See Comment (2) below on the upper bound theorem.)

(b) The statical method

Figs. 14.6-4(e), (f) and (g) show three bending moment diagrams. Each diagram has been so drawn that the moment ordinate is exactly equal to the plastic moment$M_{P}$ at two sections. In Figs, (e) and (f), α1 α2 b is the moment diagram due to the redundant moment $M_{P}$ at A, and α1c2d2b is the simple-span moment diagram due to the external loads P and 2P. In Fig. (g) the moment Mp at section C has been selected as the redundant; that is, the triangle a1a2c2bc1a1 is the redundant moment diagram (The reader should verify this. Hint: the moment Mp at C produces shear forces.), and a1a3c1d2b is the moment diagram for the loads P and 2P acting on the beam with a hinge at C. The collapse values of P can be calculated from the geometry of the three bending moment diagrams.

The three values for P obtained by the statical method agree with those obtained by the work method. As before, we conclude that the lowest value, namely P = 5Mp/4L, is the correct one.
(1) In the statical method above, we need not have calculated the collapse value of P for all the three bending moment diagrams. A closer examination will immediately reveal that both Figs. 14.6-4(f) and (g) violate the yield condition. In Fig. 14.6-4(f), the moment ordinate d1d2 exceeds Mp, which means that the collapse mechanism in Fig. 14.6-4(c) cannot occur unless plastic hinge formation is prevented at D by strengthening the cross section there. Similarly, in Fig. 14.6-4(g), a2a3 exceeds Mp ; again, the mechanism in Fig. 14.6-4(d) cannot occur unless plastic hinge formation at A is deliberately prevented. The bending moment diagram in Fig. 14.6-4(e), on the other hand, satisfies the three conditions of mechanism (with plastic hinges at A and D), equilibrium (by the manner of its construction) and yield (since moment ordinates nowhere exceed M p). Therefore we can at once conclude from the uniqueness theorem that the corresponding collapse load is the correct one; there is in fact no need to consider any other mechanisms.

$\qquad \begin{matrix} \qquad & d_{1} d_{2}&=PL-a_{1} a_{2}/4 \\ Therefore & M_{p} &=PL-M_{p}/4 \\or & P &=5M_{p}/4L \end{matrix}$

$\qquad \begin{matrix} \qquad & c_{1}c_{2} &= PL-a_{1} a_{2} /2 \\ & M_{p} &=PL-M_{p}/2 \\Therefore & P &=3M_{p}/2L \end{matrix}$

$\qquad \begin{matrix}\\ \qquad & d_{1}d_{2}&=PL/2-c_{1}c_{2}/2 \\ & M_{p}&=PL/2-M_{p}/2 \\Therefore & P &=3M_{p}/L \end{matrix}$

(2) A fundamental theorem of plastic collapse is the upper bound theorem (sometimes called the kinematic theorem) which states that for a given structure subjected to a given loading, the magnitude of the loading which is found to correspond to any assumed collapse mechanism must be either greater than or equal to, but cannot be less than, the true collapse load. (The proof of the theorem will be given in Section 14.9.)

Therefore, in an analysis we simply compute the collapse load for each possible mechanism and accept the lowest value as the correct one, as we did in the work method above. However, sometimes there is uncertainty about the number of possible mechanisms. A statical check is then necessary: the collapse bending moment diagram is drawn for the mechanism that gives the lowest collapse load; if the moment ordinate nowhere exceeds the plastic moment $M_{P}$ then the uniqueness theorem guarantees that this mechanism will give the true collapse load. If $M_{P}$ is exceeded somewhere, then the yield condition is not satisfied, and the search for a correct collapse mechanism must continue.

(3) The upper bound theorem is often referred to as the unsafe theorem, because, interpreted in a design sense, it states that the value of the plastic moment Mp obtained on the basis of an arbitrarily assumed collapse mechanism is smaller than, or at best equal to, that actually required. Consider, for example, the beam in Fig.14.6-4(a). Suppose we have to determine the plastic moment of resistance required to carry the known loads P and 2P. As we have seen in Example 14.6-1, the correct collapse mechanism (Fig. 14.6-4(b)) gives $M_{P} — 4PL/5$. The incorrect mechanisms in Figs, (c) and (d) give Mp as 2PL/3 and PL/3 respectively.

(4) Another fundamental theorem of plastic collapse is the lower bound theorem (sometimes called the static theorem), which states that if a distribution of bending moments can be found such that the structure is in equilibrium under the external loading and such that nowhere is the plastic moment of resistance Mp exceeded, then the structure will not collapse under that loading—however ‘unlikely’ that distribution of moments may appear. The theorem is often referred to as the safe theorem. (The proof of this theorem will be given in Section 14.9).

The use of the lower bound theorem is illustrated in Example 14.6-2.

(5) For future reference the three fundamental theorems of plastic collapse are displayed together below: