The opening sections of the following output file contain details of the 11 node, 10 element mesh. The left-hand side of the girder is modeled as a pin fixed against translation. The right-hand side of the girder is supported on a roller.
The latter components of the output file contain summaries of the shear and moment envelopes, together with maximum values of response quantities.
=========================================================================== Title : DESCRIPTION OF FINITE ELEMENT MESH =========================================================================== Problem_Type: Static Analysis ======================= Profile of Problem Size ======================= Dimension of Problem = 2 Number Nodes = 11 Degrees of Freedom per node = 3 Max No Nodes Per Element = 2 Number Elements = 10 Number Element Attributes = 2 Number Loaded Nodes = 0 Number Loaded Elements = 0 ------------------------------------------------------------------------------ Node# X_coord Y_coord Tx Ty Rz ------------------------------------------------------------------------------ 1 0.0000e+00 ft 0.0000e+00 ft -1 -2 1 2 5.6700e+00 ft 0.0000e+00 ft 2 3 4 3 1.1340e+01 ft 0.0000e+00 ft 5 6 7 4 1.7010e+01 ft 0.0000e+00 ft 8 9 10 5 2.2680e+01 ft 0.0000e+00 ft 11 12 13 6 2.8350e+01 ft 0.0000e+00 ft 14 15 16 7 3.4020e+01 ft 0.0000e+00 ft 17 18 19 8 3.9690e+01 ft 0.0000e+00 ft 20 21 22 9 4.5360e+01 ft 0.0000e+00 ft 23 24 25 10 5.1030e+01 ft 0.0000e+00 ft 26 27 28 11 5.6700e+01 ft 0.0000e+00 ft 29 -3 30 -------------------------------------------------------------------- Element# Type node[1] node[2] Element_Attr_Name ---------------------------------------------------------------------- 1 FRAME_2D 1 2 girder1 2 FRAME_2D 2 3 girder1 3 FRAME_2D 3 4 girder2 4 FRAME_2D 4 5 girder2 5 FRAME_2D 5 6 girder2 6 FRAME_2D 6 7 girder2 7 FRAME_2D 7 8 girder2 8 FRAME_2D 8 9 girder2 9 FRAME_2D 9 10 girder1 10 FRAME_2D 10 11 girder1 --------------------- Element Attribute Data : --------------------- ELEMENT_ATTR No. 1 : name = "girder1" : section = "no_cover" : material = "steel" : type = FRAME_2D : gdof [0] = 1 : gdof[1] = 2 : gdof[2] = 3 : Young's Modulus = E = 2.900e+04 ksi : Yielding Stress = fy = 5.000e+01 ksi : Poisson's ratio = nu = 3.000e-01 : Density = 0.000e+00 (null) : Inertia Izz = 1.727e+04 in^4 : Area = 3.809e+02 in^2 ELEMENT_ATTR No. 2 : name = "girder2" : section = "cover" : material = "steel" : type = FRAME_2D : gdof [0] = 1 : gdof[1] = 2 : gdof[2] = 3 : Young's Modulus = E = 2.900e+04 ksi : Yielding Stress = fy = 5.000e+01 ksi : Poisson's ratio = nu = 3.000e-01 : Density = 0.000e+00 (null) : Inertia Izz = 2.570e+04 in^4 : Area = 3.809e+02 in^2 ============= End of Finite Element Mesh Description ============== MATRIX : "max_displ_dead" row/col 1 2 3 units in in rad 1 0.00000e+00 -3.44203e-01 5.69206e-18 MATRIX : "max_mom_dead" row/col 1 2 3 units lbf lbf lbf.in 1 -0.00000e+00 3.02494e+03 -4.93961e+06 2 0.00000e+00 -3.02494e+03 5.14543e+06 MATRIX : "max_sh_dead" row/col 1 2 3 units lbf lbf lbf.in 1 -0.00000e+00 2.72245e+04 -8.24290e-09 2 0.00000e+00 -2.72245e+04 1.85236e+06 MATRIX : "cover_mom_dead" row/col 1 2 3 units lbf lbf lbf.in 1 -0.00000e+00 2.11746e+04 -1.85236e+06 2 0.00000e+00 -2.11746e+04 3.29308e+06 MATRIX : "moment_dead" row/col 1 units 1 lbf.in -8.24290e-09 2 lbf.in 1.85236e+06 3 lbf.in 3.29308e+06 4 lbf.in 4.32216e+06 5 lbf.in 4.93961e+06 6 lbf.in 5.14543e+06 7 lbf.in 4.93961e+06 8 lbf.in 4.32216e+06 9 lbf.in 3.29308e+06 10 lbf.in 1.85236e+06 11 lbf.in 8.24290e-09 MATRIX : "max_displ_live" row/col 1 2 3 units in in rad 1 0.00000e+00 -6.53755e-01 1.06252e-17 MATRIX : "max_mom_live" row/col 1 2 3 units lbf lbf lbf.in 1 -0.00000e+00 3.60000e+04 -9.79776e+06 2 0.00000e+00 -3.60000e+04 1.22472e+07 MATRIX : "max_sh_live" row/col 1 2 3 units lbf lbf lbf.in 1 -0.00000e+00 7.20000e+04 0.00000e+00 2 0.00000e+00 -0.00000e+00 0.00000e+00 MATRIX : "cover_mom_live" row/col 1 2 3 units lbf lbf lbf.in 1 -0.00000e+00 5.76000e+04 -3.91910e+06 2 0.00000e+00 -5.76000e+04 7.83821e+06 MATRIX : "influ_mid_mom" row/col 1 units 1 lbf.in 0.00000e+00 2 lbf.in 2.44944e+06 3 lbf.in 4.89888e+06 4 lbf.in 7.34832e+06 5 lbf.in 9.79776e+06 6 lbf.in 1.22472e+07 7 lbf.in 9.79776e+06 8 lbf.in 7.34832e+06 9 lbf.in 4.89888e+06 10 lbf.in 2.44944e+06 11 lbf.in 0.00000e+00 MATRIX : "influ_end_sh" row/col 1 units 1 lbf 7.20000e+04 2 lbf 6.48000e+04 3 lbf 5.76000e+04 4 lbf 5.04000e+04 5 lbf 4.32000e+04 6 lbf 3.60000e+04 7 lbf 2.88000e+04 8 lbf 2.16000e+04 9 lbf 1.44000e+04 10 lbf 7.20000e+03 11 lbf 0.00000e+00 MATRIX : "envelop_mom_live" row/col 1 units 1 lbf.in 0.00000e+00 2 lbf.in 4.40899e+06 3 lbf.in 7.83821e+06 4 lbf.in 1.02876e+07 5 lbf.in 1.17573e+07 6 lbf.in 1.22472e+07 7 lbf.in 1.17573e+07 8 lbf.in 1.02876e+07 9 lbf.in 7.83821e+06 10 lbf.in 4.40899e+06 11 lbf.in 0.00000e+00 impact factor = 1.275 MATRIX : "max_displ" row/col 1 2 3 units in in rad 1 0.00000e+00 -1.17786e+00 1.92411e-17 MATRIX : "max_mom" row/col 1 2 3 units lbf lbf lbf.in 1 -0.00000e+00 4.89314e+04 -1.74335e+07 2 0.00000e+00 -4.89314e+04 2.07628e+07 MATRIX : "max_sh" row/col 1 2 3 units lbf lbf lbf.in 1 -0.00000e+00 1.19037e+05 -8.24290e-09 2 0.00000e+00 -2.72245e+04 1.85236e+06 MATRIX : "cover_mom" row/col 1 2 3 units lbf lbf lbf.in 1 -0.00000e+00 9.46249e+04 -6.84991e+06 2 0.00000e+00 -9.46249e+04 1.32882e+07 START ASD CODE CHECKING:: OK : (LL+I) deflection less than 1/800 span OK : moment stress less than 0.55*Fy OK : shear stress less than 0.33*Fy
A summary of the bridge response is contained in the moment and influence line graphs.
Since this is a simple-supported bridge, the maximum displacement and maximum bending moment will occur at the middle of the span. The maximum shear force will occur at the end support.
The final results of moment, shear and displacement are calculated according to AASHTO ASD request: Total = DL + impact * LL.
The stress due to bending is given by ; Moment stress = (M * y)/I.
Because there are two different section properties, we have to calculate not only the maximum moment stress in the middle of the span (stress2), but also the moment stress of where the section changed (stress1).
Shear stress = V /(tw * d.We assume that the shear force is carried by the girder web alone, and therefore, we only check for maximum shear at the end support.
The influence line for the bending moment at the middle of the span -- details are shown in Figure 2 -- is obtained by iteratively positioning one truck load at a finite element node, then solving for the reaction forces. A similar procedure is employed to compute the influence line of shear force at the end support -- see Figure 3.
Figure 1 shows the distribution of bending moments due to truck loadings (we note in passing that the bending momend diagram corresponds to an envelope of the moment influence lines of the truck load).
The follow array elements are used in the generation of program output:
max_displ[1][2] = the maximum displacement of the beam. max_mom[2][3] = the maximum moment. max_sh[1][2] = the maximum shear force. cover_mom[2][3] = the moment at where the bridge section changed.