## Thermal Forces, Movements, and Point of Zero Thermal Movement

## Thermal Forces, Movements, and Point of Zero Thermal Movement

(OP)

Hi all, I'm struggling to find guidance relating to typical assumptions made for thermal force and movement calculations and was hoping to find some references or thoughts/reassurances as they relate to:

1. Design displacements for bearings

2. Design displacements for joints

3. Displacements/forces at piers

The thing I'm having the most difficult time conceptualizing is the interplay of the deck behavior, bearing fixity, and pier stiffness.

The specific case I'm working on is a 3 span continuous steel girder bridge with expansion joints at both ends of the bridge. We have fixed bearings at one interior pier, expansion bearings elsewhere.

1. Regarding bearing displacements it seems like it should be a case of assuming zero movement at the fixed bearing, and determining each expansion bearing displacement as a function of alpha, design temperature range, and the length from the fixed bearing to each pier (and applying factors as necessary from AASHTO).

2. Is it typically assumed that joint displacements follow the same scheme as the bearings in this situation? Or do we assume a more massive deck is going to drive the displacement (i.e. even if we have fixed bearings at one interior pier, so long as the piers have similar stiffnesses we'll end up with ~symmetrical joint movements and I should use the center of the deck as the point of zero movement)?

Perhaps I should do something in the vein of this Caltrans reference (http://www.dot.ca.gov/des/techpubs/manuals/bridge-...) I've seen linked in other threads to find a point of zero movement and then calculate displacements based on the distance from that point? I can make equivalent spring stiffnesses at each pier as a function of my pier stiffness and bearing stiffness acting in series to incorporate the difference in fixed and expansion bearings.

3. Depending on how 1. and 2. are handled, this might just be calculating an equivalent force from my anticipated bearing movement and bearing stiffness at expansion bearings (and ~0 temperature force at the fixed bearing). Or it might be that I find displacements at each pier based on the Caltrans method and apply an equivalent force to cause those displacements.

Thanks in advance for any help!

1. Design displacements for bearings

2. Design displacements for joints

3. Displacements/forces at piers

The thing I'm having the most difficult time conceptualizing is the interplay of the deck behavior, bearing fixity, and pier stiffness.

The specific case I'm working on is a 3 span continuous steel girder bridge with expansion joints at both ends of the bridge. We have fixed bearings at one interior pier, expansion bearings elsewhere.

1. Regarding bearing displacements it seems like it should be a case of assuming zero movement at the fixed bearing, and determining each expansion bearing displacement as a function of alpha, design temperature range, and the length from the fixed bearing to each pier (and applying factors as necessary from AASHTO).

2. Is it typically assumed that joint displacements follow the same scheme as the bearings in this situation? Or do we assume a more massive deck is going to drive the displacement (i.e. even if we have fixed bearings at one interior pier, so long as the piers have similar stiffnesses we'll end up with ~symmetrical joint movements and I should use the center of the deck as the point of zero movement)?

Perhaps I should do something in the vein of this Caltrans reference (http://www.dot.ca.gov/des/techpubs/manuals/bridge-...) I've seen linked in other threads to find a point of zero movement and then calculate displacements based on the distance from that point? I can make equivalent spring stiffnesses at each pier as a function of my pier stiffness and bearing stiffness acting in series to incorporate the difference in fixed and expansion bearings.

3. Depending on how 1. and 2. are handled, this might just be calculating an equivalent force from my anticipated bearing movement and bearing stiffness at expansion bearings (and ~0 temperature force at the fixed bearing). Or it might be that I find displacements at each pier based on the Caltrans method and apply an equivalent force to cause those displacements.

Thanks in advance for any help!

## RE: Thermal Forces, Movements, and Point of Zero Thermal Movement

If you're using elastomeric bearings, assuming zero movement of the piers will give you a conservative estimation of the force applied. How conservative depends on the relative stiffness of the pier and bearings. For a short, stocky pier and bearings with low shear stiffness, it's only slightly conservative. For a tall, slender pier, it may be very conservative to assume the piers don't move. The total movement is the sum of the pier bending and shear deformation in the bearing, under equal force. So the force is equal to movement divided by the sum of 1 / Kp + 1 / Kb, where Kp is the bending stiffness of the pier (3EI / L^3) and Kb is the shear stiffness of the bearings, which is the shear modulus of the elastomer (G) multiplied by total plan area of all the bearings.

For relatively stiff piers, you can assume zero movement at the fixed pier, sum the forces due to thermal expansion at the other substructures, and apply the net differential of the forces to the fixed pier. If you wish to refine it, you can use that as a starting point to iterate the final forces including deflection of the fixed pier (1 / Kb = 0 for the fixed pier).

It looks like the CalTrans method is set up for all fixed bearings at the piers, so it would takes significant adaptation to use it for your application.

If you do calculate the point of zero movement, it should end up somewhere between the centerline of the bridge and the fixed pier.

I hope that helps.

## RE: Thermal Forces, Movements, and Point of Zero Thermal Movement

When you say "we typically assume the expansion of the steel girders to be the drive the movement, unrestrained by the bearings" are you essentially using the center of your bridge length, independent of the bearing fixity? So, if you have a bridge with spans L1, L2, and L1 (end span lengths are equal) -- your interior piers will be assumed to deflect as a function of L2/2, and end piers as a function of (L1+L2/2)? Then I would calculate my pier forces as a function of displacement and their stiffness. And using those pier forces I calculate, I find also the bearing movements as a function of their stiffnesses? And the joint movements are just going to also be a function of alpha, temperature, and (L1+L2/2)?

I did look into modifying the Caltrans method yesterday and came up with something that made sense to me -- the main differences are that I integrate the bearing stiffnesses to find effective stiffnesses at each pier and use 3EI/L³ for fixed-pinned columns (I have hammerhead piers and 6 bearings at each so I just had K_eff = 1/(1/Kp+1/(6Kb)), where I set Kb to be very large for the fixed pier and the bearing shear stiffness for the others. And I did indeed find the point of zero movement to be between the fixed pier and bridge centerline.

## RE: Thermal Forces, Movements, and Point of Zero Thermal Movement

No, I was just saying that the axial forces applied by the bearings (or anything else, such as soil pressure on the abutments) is not sufficient to restrain the thermal expansion of the girders in any measurable way.

"So, if you have a bridge with spans L1, L2, and L1 (end span lengths are equal) -- your interior piers will be assumed to deflect as a function of L2/2, and end piers as a function of (L1+L2/2)?"

Only if the overall restraint at the supports is equal on each side. The final position of the superstructure after expansion (or contraction) must satisfy equilibrium of the forces applied at each substructure. With a fixed bearing at one interior pier, your bridge will not expand equally about the centerline because the fixed pier (presumably) has a higher stiffness than the other interior pier, so it will apply the same force with less displacement than the other interior pier.

Your modification of the CalTrans method looks right to me. I didn't look very closely at it, so I wasn't sure if it would be that easy, but it appears it was. It's essentially a method using the principle of equilibrium I mentioned.

Just curious, if you're using bearings with elastic shear deformation characteristics (elastomeric pads?), why use a fixed bearing at one pier instead of all expansion bearings?

## RE: Thermal Forces, Movements, and Point of Zero Thermal Movement

I think the fixed bearing is used to mostly provide clear restraint of the braking force, longitudinal components of wind forces, and the like. Although I've also seen the way you're suggesting as well, with expansion bearings everywhere, and just assuming those forces are either equally distributed or distributed as a function of their tributary lengths.

## RE: Thermal Forces, Movements, and Point of Zero Thermal Movement

The other bearings shall be aligned at radials from the point of fixity. That’s simple solution.

More advanced design could include making two center piers fixed and design accordingly for thermal stresses. The details of the design will be depended on the type of bearing used – for the narrower bridges typical fixed elastomeric bearings could accommodate some transverse expansion.

The design approach with the assumption that the main steel beams are driving the expansion is a common mistake – I have seen a lot of bridges with the shear keys misplaced by transverse expansion, or even with the cracks separating flange from the web at the outer stringers.

## RE: Thermal Forces, Movements, and Point of Zero Thermal Movement

I consider it a possibly conservative assumption. It's possible the concrete deck moderates the thermal expansion and contraction of the steel girders, but we just use the thermal expansion coefficient for steel because it's higher.

For the configurations we typically use, transverse thermal expansion isn't an issue. The elastomeric pads presumably deform due to the expansion/contraction of the the superstructure, but since all the bearings are on the same bent cap or pier, the net force on the pier shaft or columns is zero. The only effect is a negligible axial load on the cap.

"...with the cracks separating flange from the web at the outer stringers."

We haven't seen any failures due to shear from differential expansion of the deck and steel diaphragms. The difference between the expansion of the concrete deck and the expansion of the steel diaphragms over 20 to 40 feet (typical distance from the center of the bridge to the outer girder for our bridges) is theoretically less than a thousandth of an inch. If you've seen failures at a flange to web weld it was likely due to something else.