Shear lag definition

Theoretical background

As a well known fact, shear deformability in plate-girder and box-girder flanges leads to a non uniform distribution of stresses along the flange plate width. Due to bending, the longitudinal strain in the flanges and the web are equal at their connection point. The strain in the flanges, however, decreases with the distance to the web. Wide flange plates therefore elude from acting as full flange in compression or tension.

To account for this effect the flange width is often reduced and a plain strain distribution assumed.

Areas that are assumed not participating in bending stresses are called 'shear lag' areas of the cross section.

Implementation in RM Bridge

Shear lag areas may be defined in the Modeler (GP). Depending on the bending axis, shear lag areas may be defined for bending about the y axis (red,y) and for bending about the z axis (red,z) independently.

RM Bridge offers 2 possibilities for considering these shear lag definitions in the actual analysis:

1) Consideration only for stress calculation

For stiffness calculation, shear lag is not considered and the full cross section is used for calculation of cross-section area, the center of gravity (CG) and both second moments of inertia: $A^{f u l l}$ , $J_{y}^{f u l l}$ , $J_{z}^{f u l l}$ . All structural analysis calculations are performed with these values and are related to the corresponding center of gravity.

For stress calculation, these cross-section values are corrected to account for shear lag due to bending about both axes individually. Stress calculation is performed on basis of a general plain strain distribution in RM Bridge and given in principle in Eq.1.

σ = \frac{N}{A^{f u l l}} - \frac{M_{y}}{W_{y}^{r e d, y}} + \frac{M_{z}}{W_{z}^{r e d, z}}

(Eq. 1)

2) Consideration for stiffness and stress calculation

For stiffness calculation all areas of the cross-section finite element mesh defined as shear lag areas (for bending about both, or either the y or the z axis) are removed. Cross-section area, the center of gravity (CG) and both second moments of inertia are therefore calculated for the reduced cross-section: $A^{r e d}$ , $J_{y}^{r e d}$ , $J_{z}^{r e d}$ . All structural analysis calculations are performed with these reduced values and are related to the according center of gravity of the reduced section.

Self weight loads depending on the cross section area are calculated with the full area $A^{f u l l}$ and not $A^{r e d}$ .

Stress calculation is performed with the full cross-section area and with the reduced second moments of inertia. Normal stresses due to normal forces are therefore calculated on the full cross-section and normal stresses due to bending about either axis are calculated on the reduced cross-section.

σ = \frac{N}{A^{f u l l}} - \frac{M_{y}}{W_{y}^{r e d}} + \frac{M_{z}}{W_{z}^{r e d}}

(Eq.2)