The authors consider elastic buckling of an inextensible beam confined to the plane and subject to fixed end displacements, in the presence of rigid, frictionless side-walls which contain overall lateral displacements. The authors formulate the geometrically nonlinear Euler problem, derive some analytical results for special cases, and develop a numerical shooting scheme for solution. They compare these theoretical and numerical results with experiments on slender steel beams. In contrast to the simple behavior of the unconstrained problem, the authors find a rich bifurcation structure, with multiple branches and concomitent hysteresis in the overall load-displacement curves. The classical planar Euler buckling problem consists in considering a deformed arc of an initially straight, uniform road of flexural rigidity \(EI\) and length \(L\) subject to axial and lateral loads \(P\), \(F\), and moment \(N\) at the end \(S=0\), where \(S\) measures arc length; taking moments at any point \(S\neq \)0 and referring to the Cartesian coordinate system \((x,y)\) and a force sign convention, we have \[ EI\frac{d\theta}{dS}(S)+ Py(S)+ Fx(S)-N=0,\tag{1} \] where \(\theta(S)\) denotes the slope at the point \(S\) and clockwise moments are positive. The position \((x(S),y(S))\) at \(S\) is given by \[ x(S)= \int_0^S\cos \theta(\sigma)d\sigma, \qquad y(S)= \int_0^S\sin \theta(\sigma)d\sigma.\tag{2} \] Differentiating (1), using (2) and defining non-dimensional loads and moment via \[ \lambda=\frac{L^2P}{EI}, \qquad \mu=\frac{L^2F}{EI}, \qquad \nu=\frac{LN}{EI} \] we obtain \[ \theta''+\lambda\sin\theta+ \mu\cos\theta=0, \] augmented by the initial conditions \(\theta(0)= \theta_0\), \(\theta'(0)=\nu\). The initial value problem has a unique smooth solution.