Question 14.2: Design diagram of a pinned-clamped beam is presented in Fig.......

It is obvious that the first plastic hinge appears at the clamped support B; corresponding plastic moment is M_y . Now we have simply supported beam AB (this design diagram is not shown), so we can increase the load q until the second plastic hinge appears. Location of this plastic hinge will coincide with position of maximum bending moment of simply supported beam subjected to plastic moment at the support B and given load q.

For this beam, the general expressions for shear and bending moment at any section  x are

\begin{aligned}& Q(x)=R_A-q x=\left(\frac{q l}{2}-\frac{M_B}{l}\right)-q x \\& M(x)=R_A x-\frac{q x^2}{2}=\left(\frac{q l}{2}-\frac{M_B}{l}\right) x-\frac{q x^2}{2}\end{aligned}

In these expressions, the moment M_B  at the support B equals to plastic moment M_y . The maximum moment occurs at the point where Q(x)=0. This condition leads to x_0=(l / 2)-\left(M_B / q l\right) . Corresponding bending moment equals

M_{\max }\left(x_0\right)=\left(\frac{q l}{2}-\frac{M_B}{l}\right) \cdot\left(\frac{l}{2}-\frac{M_B}{q l}\right)-\frac{q}{2}\left(\frac{l}{2}-\frac{M_B}{q l}\right)^2

The limit condition becomes when this moment and the moment at the support B will be equal to M_y , i.e., M_{\max }=M_y . This condition leads to the following equation

M_y^2-3 M_y q l^2+\frac{q^2 l^4}{4}=0

If we consider this equation as quadratic with respect to M_y , then solution of this equation is

M_y=\frac{q l^2}{2}(3 \pm \sqrt{8})

Minimum root is

M_y=\frac{q l^2}{2}(3-2 \sqrt{2})=0.08578 q l^2

The limit load becomes

q_{\lim }=\frac{2 M_y}{(3-2 \sqrt{2}) l^2}=11.657 \frac{M_y}{l^2}        (b)

The plastic hinge occurs at x_0=l(\sqrt{2}-1)=0.4142 l .

If we consider equation (a) as quadratic with respect to q , then solution of this equation is

q_{\max }=\frac{2 M_y}{l^2}(3+2 \sqrt{2})

This result coincides with (b).

The following example is related to the problem of determining the limit load in the case of a simple loading of a beam by lumped force P and distributed load q. Simple loading means that in the process of increasing loads the ratio between the loads P and a remains unchanged, P=k a, where k is any fixed parameter.