143 Appendix E

143

144

CALCULATION OF THE STIFFNESS OF THE BRIDGE –

z-STAGE CONNECTION

Connection:

Model:

To evaluate the stiffness of the connection it is necessary to reduce that to a simple stiffness chain, in which the stiffness of granite, bolts and plate are combined.

Data:

Given that it is not possible to know how the plate is connected to the bridge we can assume that there are 4 bolts that join them together. Suppose that the diameter of each bolt is 10mm.

In order to obtain a reasonable model of the connection, it is necessary to make some assumptions: 1. Plate is made of steel (E)

2. ABEARING_PLATE = 4 ABOLT 3. ABEARING_GRANITE = 10 ABOLT 4. hPLATE = 0.1 LBOLT

5. hGRANITE = 0.2 LBOLT 6. LBOLT = 0.5 bridge depth The values known are:

Connection bridge plate 10A 0.1L 0.2L bolts Stiffness KGRANI TE KBOLT KPL ATE KSPRING

145 Young modulus of steel E=210GPa

Young modulus of granite (EGR) = 52.5GPa Bridge depth l=150mm

Area of the bolt

2 2 78.54 4 D A=

π

Remind that there are two different ways to combine springs:

The equivalent stiffness of the simplified chain is:

(

)

(

(

)

)

1 1 1 PLATE BOLT GRANITE

SPRING GRANITE PLATE BOLT GRANITE PLATE BOLT

K K K K K K K K K K + + = + = + ⋅ + therefore:

(

)

GRANITE PLATE BOLT

SPRING

GRANITE PLATE BOLT

K K K K K K K ⋅ + = + + where: 0.25 10 0.2 4 0.1 GR GR GRANITE GR PLATE PLATE PLATE PLATE BOLT BOLT E A E A K L L E A E A K L L E A E A K L L ⋅ ⋅ = = ⋅ ⋅ = = ⋅ ⋅ = =

Replacing the values, it is obtained the connection’s stiffness as follows: Serial: Parallel: 1 2 1 1 1 1 ... REPLACE n K = K +K + +K 1 2 ... REPLACE n K =K +K + +K

146 2106.6 SPRING N K m

µ