# Finite Element Analysis of a Two-Span Highway Bridge

### PROBLEM DESCRIPTION

In this example, we will use ALADDIN for the three-dimensional finite element analysis of a two-span highway bridge structure. We will compute:

• The influence line of mid-span displacements caused by a 1000 kip concentrated live load moving along one of the outer bridge girders.

### DESCRIPTION OF HIGHWAY BRIDGE

A plan and front elevation view of the highway bridge system is shown in Figures 1 and 2.

#### Figure 2 : Cross Section View

The bridge has two spans, each 100 ft long. The width of the bridge is 16 ft - 8 inches. The left-hand side of the bridge is a hinged support, and the right-hand side, is supported on a roller.

The bridge is constructed from one steel and one concrete material type. The structural steel has Fy = 50 ksi, Poisson's ratio v = 0.3, and E = 29,000 ksi. For the concrete slab, fc = 4000 psi, Poisson's ratio v = 0.3, Ec = 3625 ksi, and unit weight = 150 pcf. The W36x170 steel section has the following properties -- d = 36.17 in, bf = 12.03 in, tf = 1.1 in, tw = 0.68 in, and weight = 170 lbf/ft. The concrete slab has thickness 7 in.

### FINITE ELEMENT MODELING

Figures 3 and 4 are plan and cross section views of the finite element mesh, respectively.

#### Figure 4 : Cross Section View of Finite Element Mesh

The finite element model has 399 nodes and 440 shell elements. After the boundary conditions are applied, the model has 2374 d.o.f.

Figure 5 is a MATLAB plot of the computed bridge deck deflections due to dead loads alone.

#### Figure 5 : Bridge Deck Deflections

As expected, the vertical deck deflections are symmetric about the center support, and the axis of symmetry in the bridge's longitudinal direction.

Figure 6 shows the plan and elevation views of the bridge with the 1000 kip moving live load.

#### Figure 6 : Bridge Deck Deflections

The influence diagram is computed by systematically moving the point load along finite element nodes over-lying the outer bridge girder.

```/* [d] : Compute Influence Lines for Moving Live Loads */

print "\n*** STATIC ANALYSIS PROBLEM \n\n";

Fx = 0 lbf;    Fy = 0 lbf;    Fz = -1000.0 kips;
Mx = 0 lbf*in; My = 0 lbf*in; Mz = 0 lbf*in;

nodeno1 = 97;
nodeno2 = 103;
step   = div_L*2+1;
print "node no 1 =",nodeno1,"\n";
print "node no 2 =",nodeno2,"\n";
print "Fz        =",Fz,"\n";
print "step      =",step,"\n";

influ_line1 = Zero([ step , 1 ]);  /* array for influence line 1 */
influ_line2 = Zero([ step , 1 ]);  /* array for influence line 2 */

lu = Decompose (stiff);
for( i=1 ; i<=step ; i=i+1 ) {

displ = LUDecomposition( stiff, eload );

node_displ_1 = GetDispl( [nodeno1], displ );
node_displ_2 = GetDispl( [nodeno2], displ );

influ_line1[i][1] = node_displ_1[1][3];
influ_line2[i][1] = node_displ_2[1][3];

}

PrintMatrix(influ_line1);
PrintMatrix(influ_line2);
```

You should notice that our moving load analysis begins with the decomposition of the stiffness matrix into a product of lower and upper triangular forms -- this step requires O(n^3) computational work. Then the bridge deck displacements are computed with repeated applications of forward and backward substitution. Each set of displacements is obtained with only O(n^2) computational work.

The influence lines of mid-span displacement are stored in the arrays influ_line1 and influ_line2, one for each of the bridge girders.

#### Figure 9 : Influence line of Midspan Displacement

Figure 9 is the influence line of midspan displacement under the girder along which the point live load is moving.