2019 CSCE Annual Conference - Laval (Greater Montreal) Conference
Dr. Pierre Leger
Three-dimensional structural analysis methods for deep plain concrete spillway piers subjected 3D loads (P-Mx-My-Vx-Vy-T) can be divided into three categories: (i) the gravity method; (ii) 3D fiber elements; and (iii) 3D Finite Element method (FEM). The gravity method (GM) is used frequently in engineering practice. It is based on the Euler – Bernoulli beam theories with the assumption that cross sections remain plane after deformation. However, nonlinear normal and shear stress distributions induced by warping is not captured. Element warping can be captured by using 3D FEM using commercial software such as ABAQUS. Yet, 3D FEM requires significant resources and complex post-processing to compute classical engineering stability indicators such as (i) the sliding safety factor, (ii) the position of the force resultant, (iii) the cracked area, and (iv) the maximum compressive stresses. 3D fiber elements use conventional beam theory input parameters such as cross section area (A), moments of inertia (Ix, Iy, Ixy), shear sectional area (Ax, Ay, Axy) and torsional constants (J, G). Higher order beam theories leads to more precise results than the GM but with less complexity than 3D FEM in terms of engineering resources and result interpretation to take decision about the adequacy of stability indicators as compared to legal code requirements.
This paper presents a new higher order 3D fiber element using beam theory leading to a 18x18-stiffness matrix including shear and torsion warping. This element has two nodes and 9 degrees of freedom (DOFs) per node; 6 DOFs for normal displacements, and 2 additional DOFs for shear warping displacements in X and Y, and 1 DOF for torsional warping displacement (around Z). The element stiffness matrix is numerically integrated using a mixed isoparametric formulation: (i) the flexibility method is used for flexural and shear influence coefficients; (ii) the displacement method is used for torsional influence coefficients (Saint–Venant and bimoment). The element (fibers) cross sections along the beam are first discretized using 2D FEM. The element stiffness matrix is then computed from Gauss integration of cross sectional stiffness coefficients.
This new 3D fiber element element is implemented in a MATLAB code. Validations and verifications are performed using first a simple example of a plain concrete deep beam comparing simulation results to the exact solutions from elasticity theory. A more geometrically complex concrete deep pier of an existing spillway is then analyzed using ABAQUS (3D FEM) to validate the proposed fiber element model.