axial force
The steel plate girder shown in Fig. 6.2-69 is restrained against lateral torsional buckling and is initially assumed to be a Class 2 cross-section under bending and axial force. The girder is part of a single-span integral bridge and receives a compressive thrust from the abutments of 10 600 kN applied at the level of the plastic neutral axis for bending moment alone. The maximum sagging bending moment that the section can withstand in conjunction with the axial force is calculated and a check is made to ensure that the cross-section remains Class 2. All plates are grade S355 to EN 10025 and the yield strengths for different plate thicknesses are to be taken from 3-1-1/Table 3.1. (Note that the UK National Annex requires the values from EN 10025 to be used.)
400 500 45 40 3-1-1/Table 3.1 – fy = 355 MPa 3-1-1/Table 3.1 – fy = 335 MPa 1225 40
Fig. 6.2-69. Plate girder for Worked Example 6.2-16
The compression flange is first classified using 3-1-1/Table 5.2. Conservatively ignoring the web-to-flange welds, the flange outstand c¼ ð400 40Þ=2 ¼ 180 mm. c=t ¼ 180=40¼ 4:5. 9" ¼ 7:3 4:5, so the flange is Class 1.
2fyd (comp.)
fyd (comp.) fyd (comp.)
fyd (tens.) fyd (tens.)
504.6 mm (a) (b) (c) a Plastic neutral axis Equal force
axis Zone carryingaxial force
Fig. 6.2-70. Stress block for Worked Example 6.2-16: (a) stress block for bending; (b) stress due to axial force; (c) final stress block
The plastic section properties of the girder are found to be as follows.
Equal force axis¼ 504.6 mm from bottom of web. This is the location of the plastic neutral axis for bending alone.
Plastic moment of resistanceðMpl;RdÞ ¼ 12 370 kNm.
Figure 6.2-70 shows the stress distribution under combined bending and axial force. The depth ‘a’ is first calculated.
Assuming the plastic neutral axis occurs in the web, force balance gives: 10 600 103¼ a 40 2 355=1:0 so a ¼ 373.2 mm < 504:6 mm
The assumption is therefore correct – the plastic neutral axis occurs in the web.
Therefore, the plastic moment of resistance about the equal force axis of the section resisting axial forceðM2fydÞ ¼ 373:2 40 2 355 373:2=2 1 106¼ 1977 kNm.
Therefore the resulting plastic moment of resistance in the presence of axial force MN;Rd¼ Mpl;Rd M2fyd¼ 12 370 1977 ¼ 10 393 kNm.
The section can withstand a maximum sagging moment of 10 393 kNm in the presence of a 10 600 kN axial force (applied at the level of the plastic neutral axis for bending moment alone). This method would need modification if the yield stress was different in web and flange and the neutral axis was located in the flange. It would be simplest to use the smallest value of yield stress throughout.
It is now checked that the cross-section is still Class 2 in the presence of the axial force. 3-1-1/Table 5.2 – Web is ‘Part subject to bending and compression.’
>0:5 (by inspection) c¼ depth of web ¼ 1140 mm
c¼ depth of web in compression ¼ 1140 504:6 þ 373:2 ¼ 1008:6 mm Therefore, ¼ 1008:6=1140 ¼ 0:885
For the web to be classified as Class 2, c=t 456"=ð13 1Þ where: t¼ thickness of web ¼ 40 mm "¼ 0:81 (3-1-1/Table 5.2) c t¼ 1140 40 ¼ 28:5 and 456" 13 1¼ 456 0:81 13 0:885 1¼ 35:2 > 28:5
Therefore the web is still Class 2 despite the compression forces. It will be further noted that this section would still be compact if the whole web depth was in compression.
6.2.11. Bending, shear and axial force
The bending resistance of cross-sections resisting combined bending, shear and axial force may be reduced by both the axial force and shear components of the loading. 3-1-1/clause 6.2.10(1) requires this effect to be considered. The method of interaction again depends on the class of the cross-section and whether or not the shear resistance is limited by shear buckling.
This section of the guide is split into two sub-sections as follows:
. Sections not susceptible to shear buckling Section 6.2.11.1
. Sections susceptible to shear buckling Section 6.2.11.2
6.2.11.1. Sections not susceptible to shear buckling
6.2.11.1.1. Class 1 and 2 cross-sections
Where there is no shear buckling, the effect of shear on cross-section resistance need only be considered if the shear force exceeds 50% of the design plastic resistance – 3-1-1/clause 6.2.10(2)refers. The first step in checking the cross-section is to establish the reduced web yield strength, or thickness, caused by shear as discussed in section 6.2.9.1.1 of this guide – 3-1-1/clause 6.2.10(3) refers. The resulting reduced section is then checked for combined bending and axial force using plastic section design in accordance with section 6.2.10.1 of this guide. If the section is not symmetric, the plastic neutral axis will shift when the reduction in web strength is made. The comments made in section 6.2.9.1.1 regarding not reclassifying the web to 3-1-1/Table 5.2 after modifying the cross-section for shear apply here also; the section classification is checked first under the bending moment and axial force before any reduction is made to the web strength for shear.
An alternative simpler and more conservative approach is to use a linear interaction as permitted by 3-1-1/clause 6.2.1(7): NEd Nv;Rd þ My;Ed Mv;y;Rd þ Mz;Ed Mv;z;Rd 1:0 (D6.2-58)
where Mv;y;Rd and Mv;z;Rd are the reduced resistance moments allowing for shear, but not
axial force, about the y–y and z–z axes respectively. Nv;Rd is the axial resistance based on
a cross-section with reductions for shear. The comments on section classification under axial force and bending made in section 6.2.10.1 of this guide apply when using this linear interaction.
6.2.11.1.2. Class 3 cross-sections
The procedure for treating Class 3 cross-sections is slightly different to that for Class 1 and 2 cross-sections since elastic section design has to be used for combinations of bending and axial force. There are three possibilities for doing the check:
(i) Establish the reduced web yield strength or thickness caused by shear (if the shear force exceeds 50% of the plastic resistance) as for Class 1 and 2 cross-sections. The resulting reduced cross-section is then checked for combined bending and axial force using elastic section design in accordance with section 6.2.10.2 of this guide. It makes most sense to reduce the web thickness rather than its yield strength or the beam resistance will be governed by yielding of the web at this reduced yield stress.
(ii) Use the interaction given in 3-1-5/clause 7.1 which is intended for cases with shear buckling as discussed in section 6.2.11.2.2 of this guide.
(iii) Use the interaction of equation (D6.2-58). The use of plastic section properties in deter- mining Mv;Rd (as long as the resulting resistance does not exceed the elastic bending
resistance) is discussed in section 6.2.9.1.2. It is necessary to use this method to avoid a discontinuity with the moment resistance with shear alone if this has been determined to EC3-1-1 clause 6.2.8 as discussed in section 6.2.9.1.2 of this guide.
The three methods are illustrated in Worked Example 6.2-18.
Once again, if the section is not symmetric, the neutral axis will shift when the reduction in web thickness for shear is made. There is no need for the section classification to be rechecked for this shift. Methods (ii) and (iii) are significantly more economical.
3-1-1/clause 6.2.10(1) 3-1-1/clause 6.2.10(2) 3-1-1/clause 6.2.10(3)
6.2.11.1.3. Class 4 cross-sections
Class 4 cross-sections have to be dealt with using one of two possible methods given in EN 1993-1-5. These are discussed in section 6.2.11.2.3 below as the procedure is the same whether or not there is shear buckling.