Section cut, continuous box girder

Results.

Problem. In a two-span continuous box girder the support reactions depend on the stiffness of the girder and cannot be found from statics alone. The section cut gives the shear force, the bending moment and the torsional moment in the spans and on both sides of the interior support.

Reference document. The value of the section is checked with the free-body equilibrium of the part to its left. This equilibrium uses the model's own support reactions.

The support reactions $R_i$ to the left of the section and the load $w$ uniform along the girder give these values at the section $x = a$. The torsional moment follows from the force couples of the bearings in the same part.

Numerical model. The girder has two 30 m spans and is built from shell elements with the box section of the first bridge. There is a 0.50 m diaphragm on every support line and the mesh is 0.50 m quadrilateral elements.

Two-span continuous box girder, 2 × 30 m#

Closed form. Static equilibrium with the model's own support reactions

Note. The bearings are at the nodes where the bottom slab meets the webs, and the two bearings on every support line are held vertically. Only the interior support is held along the girder. Q is a 10 kPa load on a 2.5 m wide strip of the first span only, 1.75 m from the girder axis. The sections are at x = 1, 11.5, 29, 31 and 45 m. The sections at 29 and 31 m lie on either side of the interior support, and the difference in shear between them comes from the interior reaction.

Comparison with continuous beam theory#

Compared model. Continuous beam theory (Euler-Bernoulli bending, St. Venant torsion)

Note. Beam theory leaves out the shear deformation of the box and the rotation of the section on a support line. Under G the end reaction is 0.50% larger in the model, which is why the shear differs by the same amount at every section (5.06 kN). With shear deformation taken into account, beam theory gives an end reaction 0.40% larger. On a support line the section is held only at its two bottom corners, so the diaphragm deforms and the top of the section rotates. At the interior support this rotation is 8% of the one at midspan. Under Q the torsion of the first span is therefore shared between its ends as 640 and 672 kNm, and the second span takes 14.7 kNm. Theory gives 656, 656 and zero.

Comparison with SAP2000#

Independent calculation. SAP2000 27.1.0, the same model, Section Cut by SAP2000's own group definition

Note. The SAP2000 model is built with the same nodes and elements, the shells are Shell-Thick and SAP2000 does not mesh them further. The Section Cut uses SAP2000's own definition, a group of the summed elements and the section nodes, and the moment is taken about the same point. Shear and moment agree to better than one in a thousand. How the torsion is shared between the supports depends on the shell element of each program, so the torsion differs by 0.4% in the first span, and the torsion passed to the second span is 14.7 kNm in MindFEM and 16.3 kNm in SAP2000.

Model files#

The model of the SAP2000 comparison can be downloaded. It opens in SAP2000 27 and later with File > Import > SAP2000 .s2k Text File and carries the Section Cut definitions.

• Two-span continuous box girder (.s2k)