## 2 In-Plane Bending of Cantilever Beam  
The first benchmark concerns the bending problem of a cantilever beam with a rectangular cross-section. The configuration is reduced to a two-dimensional domain, and the problem is actually determined as an in-plane bending of a flat membrane; hence, the plane stress state is considered. Its analytical solution is given, for example, in [23, 24]. This study focuses on the convergence examination for the numerical solution. Figure 1 depicts the specimen together with corresponding boundary conditions and additionally a discretization introduced by the medium mesh of quadrilateral (square) finite elements. On the left, a parabolic tangential load \(q(y)\) is defined with a maximum value of \(q_{\rm max}=500\) kN/m for point 0, i.e., for the origin of the coordinate system where \([x,y]=[0,0]\). After integration of the load, the resultant \(P=40\) kN. On the right, a fixed (clamped) edge is set; however, according to the theoretical solution, two options of beam constraints can be regarded; see details, e.g., in [12]. For the first type, the effect of shear stresses is omitted, and it conforms with the Bernoulli-Euler theory. In the second case, the shear stresses are taken into account, so the Timoshenko-Ehrenfest theory [27, 28] is valid. The data introduced in the computations are as follows: [PERSON]'s modulus is \(E=200\) GPa, and [PERSON]'s ratio is \(\
u=0.25\). The definition of the shear modulus is also needed according to the Timoshenko-Ehrenfest theory; hence, \(G=80\) GPa. The length is \(L=48\) cm, the height 12 cm corresponds to \(2\,c\) in Figure 1, and the cantilever thickness is \(b=1\) cm. The moment of inertia for the beam cross-section is \(J_{c}=b\,(2c)^{3}/12=144\) cm\({}^{4}\). In this test, some characteristic quantities are monitored at points indicated in Figure 1. The definition of horizontal displacement \(u(x,y)\) as well as vertical displacement \(v(x,y)\), called also the beam deflection, is governed by the two aforementioned theories; see also [23]. According to the Bernoulli-Euler theory, on the clamped edge at point \([x,y]=[L,0]\), it is assumed that \(\partial u/\partial y\
eq 0\) and \(\partial v/\partial x=0\), so the influence of shear stresses on the beam deflection is neglected. Based on the solution for this theory, the horizontal displacement is expressed as the function:  
\[u^{\rm B-E}(x,y)=-\frac{P}{2 EJ_{c}}\,x^{2}\,y+\left(-\frac{\
u P}{6 EJ_{c}}+ \frac{P}{6 GJ_{c}}\right)y^{3}+\left(\frac{P\,L^{2}}{2 EJ_{c}}-\frac{P\,c^{2}}{ 2 GJ_{c}}\right)y \tag{1}\]  
The variable name is supplemented by upper index B-E. This displacement differs in sign depending on the \(y\)-coordinate, specifying points A and B, i.e., the corners on the left-hand side:  
\[u^{\rm B-E}(0,+c)=-u^{\rm B-E}(0,-c) \tag{2}\]  
Figure 1: Model of cantilever beam with parabolic load and its resultant on left-hand side, together with characteristic points and medium mesh (here, FE size is 3 cm). Thereafter, in the computations, only positive horizontal translation given at point B is observed and referenced to the value, which for the above data equals:  
\[u^{\text{B-E}}(0,+c)=-\frac{vP\,c^{3}}{6 EJ_{c}}+\frac{P\,L^{2}\,c}{2 EJ_{c}}- \frac{P\,c^{3}}{3 GJ_{c}}=0.09337\ [\text{cm}] \tag{3}\]  
The beam deflection, determined by the Bernoulli-Euler theory, is  
\[v^{\text{B-E}}(x,y)=\frac{v\,P}{2 EJ_{c}}x\,y^{2}+\frac{P}{6 EJ_{c}}x^{3}-\frac{ P\,L^{2}}{2 EJ_{c}}x+\frac{P\,L^{3}}{3 EJ_{c}} \tag{4}\]  
The maximum deflection given at point 0 is then equal to  
\[v^{\text{B-E}}(0,0)=\frac{P\,L^{3}}{3 EJ_{c}}=0.512\ [\text{cm}] \tag{5}\]  
When the Timoshenko-Ehrenfest theory is taken into account, the conditions \(\partial u/\partial y=0\) and \(\partial v/\partial x\
eq 0\) hold for the clamped edge at point \([x,y]=[L,0]\), and the impact of shear stresses on the beam deflection is accounted for.