Test 1.50. Two-rib beam subjected to uniformly distributed loads applied in the plane of the ribs

Objective

To explore the distribution of normal stresses in a two-rib beam subjected to uniformly distributed loads applied in the plane of the ribs.

Reference

A.V.Alexandrov, B.Y.Lashchenikov, N.N.Shaposhnikov. Structural mechanics. Thin-walled spatial systems. — Moscow: Stroyizdat, 1983.

Problem statement

A two-rib beam is simply supported by end diaphragms that are perfectly rigid in their plane and perfectly flexible out of their plane. The beam is subjected to uniformly distributed line loads q applied along the ribs in their plan.

To determine the longitudinal normal stress σ_xi at cross‑sectional points i = 1, 4, 5, and 6 at half‑span (l/2) and quarter‑span (l/4), considering the assumptions adopted in deriving the analytical solution:

1. Bending deformations of the beam components out of their plane are neglected. 
2. No transverse displacements in the horizontal plane are assumed at the connections between the ribs and the flange.
3. Differences in stresses between the beam components at the rib-to-flange connections are neglected.

Design model

The design model is model type 5; 6 DOF per node. The FE mesh is divided with a step of 0.25 m in the transverse direction of the beam and 0.2453125 m in the longitudinal direction. Internal forces are to be determined along the X-axis of the global coordinate system.

Geometry

Rib and flange thickness δ = 0,1 m
Rib height b = 1 m
Distance between the ribs 2*b = 2 m
Flange width 4*b = 4 m
Beam length l = 7,85*b = 7,85 m

Material properties

Modulus of elasticity Е = 3*10⁷ kPa
Poisson's ratio ν=0,15

Boundary conditions

The boundary conditions are provided by applying restraints:
- in the Y degree of freedom along the ribs;
- in the Y and Z degrees of freedom along the edges parallel to the Y-axis;
- in the X, Y, and Z degrees of freedom at the origin.

Loads

Uniformly distributed line load applied along the ribs q = 10 kN/m.

Output data

Analytical solution

For an analytical solution based on seven harmonics of the unknown generalized displacements, the longitudinal normal stresses σ_xi (kN/m²) in the beam components at cross-sectional points i = 1, 4, 5, and 6 at half-span (l/2) and quarter-span (l/4) can be calculated using the following formulas:

Comparison of calculation results

Without additional side nodes:

With additional side nodes: