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Chapter 3 ISOLATOR DEVICES AND SYSTEMS
3.3 STEEL HYSTERETIC DAMPERS
3.3.3 Approximate Force-Displacement Loops for Steel-Beam Dampers Stress-Strain Loops and Force-Displacement Scaling Factors Stress-Strain Loops and Force-Displacement Scaling Factors
The family of force-displacement loops for a bending-beam or twisting-beam damper can be scaled on the basis of a simple model, to give a set of stress-strain curves. Approximate force-displacement loops for a wide range of steel-beam dampers can then be obtained from the scaled stress-strain curves.
Figure 3.4: Scaled stress-strain loops for Type-Tr steel-beam damper, made of hot-rolled mild steel complying with BS4360/43A. This diagram can be used to generate approximate force-displacement loops using the scale factors for the 7 types of steel-beam damper given in Table 3.1.
Table 3.2: Scaling Factors for Steel Beam Dampers
Figure 3.4 shows scaled stress-strain loops for a Type Tr steel-beam damper made of hot-rolled steel complying with BS4360/43A. Table 3.2 shows the force- and displacement-scaling factors, f and l respectively, for 7 types of damper.
The scaling factors f and l of Table 3.2 and Figure 3.4 are based on a greatly simplified but effective model of the yielding beam. The extreme-fibre strains (or ) are based on the shape which the beam would assume if it remained fully elastic. The nominal stresses or^ are related to the force scaling factor f on the assumption that they remain constant over a beam section (as they would for a rigid-plastic beam material.) The circumflex (^) is introduced to emphasise the nominal nature of the stresses and moduli derived using the uniform-stress assumption.
It can be shown that premultiplication of the scaling factor f by about 0.6 will correct to some extent for the approximation's nonvalidity. However, if such refinement is required, it is preferable to scale using the method of c(iii) and d(iii) below.
The force F and displacement X can then be obtained
where
, ˆ
6 are given by Figure 3.4, , ˆ
7 are given approximately by Figure 3.4, by letting = and /2, and where 'a' is a small correction factor for large-displacement shape changes.
For dampers of Type U, T and E respectively, values of the correction factor 'a' are:
where R and L are defined in Table 3.2.
) or ( ,
X
(3.2a)X )) a + (1 ˆ f (or X ), a + (1 ˆ f
F
2
2 (3.2b)R ), /(2 a 1
) ; R /(L+
a 2 R );
/(8 1
a -
E 22 T
2
U
(3.2c)
54
Figure 3.3(c) is an example of the effect of a positive 'a' value on the loop shapes of Figure 3.1(b). The positive aT and aE values of equation (3.2c) cause an increase in the slope of the force-displacement loop for large yield displacements of T-type and E-type dampers, in accordance with equation (3.2b). Similarly, the negative aU value causes a reduction in the loop slope for large yield displacements of U-type dampers.
The stress-strain loops of Figure 3.4 were derived from force-displacement loops for a Tr-type damper, using equations (3.2) and f(Tr) and l(Tr) values from Table 3.2. The force-displacement loops in Figure 3.4 were not corrected for beam-end effects, since these were considered typical for bending-beam dampers. Hence damper designs based on Figure 3.4 and Table 3.2 already includes typical beam-end effects. The initial stiffness of the damper is somewhat uncertain, owing to variations in end-effects and the stiffness of beam-loading arms.
When equations (3.2) are used to generate the stress-strain loops from the force-displacement loops of a Tr damper, they eliminate the large-displacement increases in nominal stresses, as is evident from a comparison of Figures 3.3(c) and 3.4. When dampers are then designed using Figure 3.4, equations (3.2) reintroduce appropriate large-displacement changes in force and stiffness.
By introducing the very rough approximation
ˆ 2
ˆ9 and using =, Figure 3.4 and Table 3.2 can be used to obtain a rough estimate of the force-displacement loops for E-type (torsional) dampers. However, it would be more accurate to generate a separate set of ˆ -
10 loopsbased on force-displacement loops for an E-type damper and equations (3.2). A representative beam section should be used, say a rectangle with B=2t, where B and t are defined in Table 3.2.
Alternatively, the method of c(iii) should be used if more accuracy is required.
Errors in Approximate Damper Loops
There are four main sources of error in the damper loops and parameters derived by the method described in c(i) above.
(1) Differences between the material properties of the hysteretic beam used to generate the stress-strain loops of Figure 3.4 and the material properties of the hysteretic beam in the prototype.
(2) End-effects and non-beam deformations. End-effects usually reduce the initial stiffness by about 50% and are particularly important for rectangular-beam type-E dampers.
(3) Alteration of loop loads, for a given displacement, by changes in the shape of the damper under large deflections. Shape changes reduce Kb2 for type-U dampers and increase Kb2 for type-T and type-E dampers. First-order corrections have been derived for the load changes due to damper shape changes. These have been used to remove large-deflection effects from the loops in Figure 3.4.
(4) Small changes in the damper loops caused by secondary forces. For example, the E- type damper is deformed by bending as well as by twisting forces. These effects have been small or moderate for all the damper proportions tested.
The inelastic interaction of primary and secondary beam strains results in a gradual progressive cycle-by-cycle change in beam shape. The beam of a U-type damper deforms progressively away from a line through the loading pins. The beams of an E-type damper deform progressively towards the axis of the loading pin. These effects were not serious in any of the dampers tested.
The method given below in c(iii) gives a more accurate procedure for generating force-displacement loops for steel-beam dampers.
Damper Loops Derived from Models of Similar Proportions
A scale-model method partially eliminates the four sources of error given above. In this method, force-displacement loops are derived for an experimental model, or damper of similar but not identical proportions as the prototype, and made of the same material. The scaling is then done in terms of the force- and displacement-scaling factors, f and l, given in Table 3.2.
If subscripts p and e are used for the 'prototype' and the 'experimental model' respectively, then, neglecting the correction factors involving 'a' of equation (3.2c)
For example, for a Uc damper, Table 3.2 gives
Section (d) below describes how the stiffness ratios and yield-point ratios can also be obtained.