Effect of Horizontal Shear Transfer on Stresses Beneath Steel Bearing Plates

In practice, the design of pot bearing steel plates is usually based on assuming that stresses beneath the plate are uniformly distributed. There is also no consideration given to the effect of shear stresses transmitted through the interface between the surfaces of the Plates and the supporting concrete abutment. However, in this analytical investigation, the behavior of pot bearing steel plates supported by concrete abutment was studied using finite elements analysis. There were no special assumptions made concerning stresses beneath the plate. Nevertheless, the interface between the surf aces of the steel plate and the concrete abutment was simulated using vertical and horizontal interface spring elements. Normal and shear stresses were permitted to be transferred through the interface with slip occurring beyond some limiting shear strenqth value. Different shear transfer conditions were induced and stresses resulting in the bearing system were studied using axisymmetric and plane strain solution methods. Results indicated that assuming uniform stress ii distribution beneath the plate is rather too conservative. However, results from different shear transfer conditions showed that the effect of shear transfer is practically insignificant for design purposes.


ABSTRACT
In practice, the design of pot bearing steel plates is usually based on assuming that stresses beneath the plate are uniformly distributed.
There is also no consideration given to the effect of shear stresses transmitted through the interface between the surfaces of the Plates and the supporting concrete abutment.
However, in this analytical investigation, the behavior of pot bearing steel plates supported by concrete abutment was studied using finite elements analysis. There were no special assumptions made concerning stresses beneath the plate. Nevertheless, the interface between the surf aces of the steel plate and the concrete abutment was simulated using vertical and horizontal interface spring elements. Normal and shear stresses were permitted to be transferred through the interface with slip occurring beyond some limiting shear strenqth value.
Different shear transfer conditions were induced and stresses resulting in the bearing system were studied using axisymmetric and plane strain solution methods.
Results indicated that assuming uniform stress ii distribution beneath the plate is rather too conserva- My parents, Mr. and Mrs. Arouri, for their love, sacrifice, encouragement, and unlimited support.
My sister Rima Arouri, and my friends Georgette and Samia Kiriaki for all the help they gave me in assembling the thesis to its final form.
Mrs. Donna Brightman for her concern and good will.  List of relative vertical and horizontal displacements between various nodes adjacent to the interface zone (inches).

Vertical interface spring elements cross-sectional areas
Horizontal interface spring elements cross-sectional areas .
Axisymmetric solution. Averaged bending stresses (Ksi) in top of the pot bearing steel plate.
Plane strain solution. Averaged bending stresses (Ksi) in top of the pot bearing steel plate.
Gauss integration points 2 and 4 Axisymmetric solution.
Averaged bending stresses (Ksi) in top of the masonry steel plate.

Plane strain solution.
Averaged bending stresses (Ksi)  Beam on elastic foundation model used in the previous analysis Bearing system finite element two-dimensional model used in the previous analysis .
Actual and assumed stress distribution in the concrete beneath the masonry plate Actual structure and finite element grid system .

Beam element
Truss element in space Possible two-dimensional finite element analysis A general two-dimensional element .
Three-dimensional solid model .       2) Equilibrium element, based on assumed stress functions; and 3) Hybrid element, which is based on both assumed displacement and stress functions. Elements (2) and (3) are used to a much lesser extent. As described in the next chapter, displacement elements are used in the modeling of the bearing system. some of the elements used in modeling the bearing system will be discussed with more depth in later chapters. u(x) ( 3. 1) e (x) where e is the first partial derivative of u(x)

Axisymmetric cylindrical model
representing nodal rotations.
Expressing (3.1) in matrix notation, As shown, the number of degrees of freedom equals the number of undetermined coefficients in (3.1).

FORMULATION OF STIFFNESS MATRIX AND LOAD MATRIX
The next step which immediately follows the formulation of the displacement function, is to find nodal strains and stresses. Those strains and stresses will be used to construct the stiffness matrix of the element and its equivalent nodal forces. A general brief description of the procedure used to formulate the element stiffness matrix will be presented in this section. The principle of minimum total potential energy will be used in the formulation.
As shown in the above section in equation ( ( 3 • 8) where {E} = strain matrix The stresses are related to the strains by the [C] matrix, which is called the elasticity or the property matrix .
Substituting (3.8) in (3.9) (3.10) Total potential energy (IT) is the sum of the strain energy (U) and the potential energy (V) , thus The strain energy Substituting (3.8) and (3.10) in (3.12) vol d vol (3.11) (3.12) ( 3. 13) The potential energy is where {u } is taken from equation (3.7) and {q } is the distributed load matrix .
substituting (3.13) and (3.15) in (3.11) The principle of stationary potential energy 27 requires that the first partial differential of the total potential energy with respect to any nodal displacement must be zero. Thus, the equilibrium condition to be satisfied is, Substituting (3.16) in (  . c ig. ( 3. 3) Truss element in snace . 30 element is neglected during deformation.
(2) Two-dimensional Elements The two-dimensional elements available in NONSAP are: 1) Plane stress elements, with the assumption that out of plane shears and stresses are negligible.
2) Plane strain elements where the out of plane strains are negligible.
A system of natural coordinates (r,s) were given to the element such that r = +l along edge (1-8-4), and r = -1 along edge (2-6-3); while s = +l along edge (2-5-1) and s = -1 along edge (3-7-4). The global coordinate is given in cartesian coordinates (y,z); however, polar coordinates could also be used. Let (u ,6 ) and (u , 6 ) represent the displace- relating (y,z) to (r,s) derivatives is: (3.25) where J is the Jacobian operator, written as  that the AISC design manual [11] recommends that the width of the bearing plate be at least 2 inches less than the width of the support, which is concrete in this case. The depth of the concrete was selected so that the stress distribution on the top of it will not be affected by the In this investigation the effect of the shear transfer will be discussed in depth in later chapters by the use of three different models which represent the three different cases of shear transfer.
Since the bearing system is symmetric about the vertical central axis, only half of each model will be analyzed to save calculation effort and computer time.
The following are the three cases of shear transfer: 1) The case of zero shear transfer. There is no shear force transferred along the contact zone between the steel masonry plate and the concrete abutment.  The element is usually described in a cylindrical coordinate system whose axis of symmetry is the (z-axis) and for which radial distances are defined by the coordinate (r) and the circumferential coordinate is described by an angle 9 as shown in fig. (4.8). However, in this investigation the axisymmetric set of elements will be selected such that (z-axis) will be the axis of and if (P) is the uniformly applied, load, then the resultant load is (R), where R = P · A but since the analysis is done by the finite element ( 4 • 2) method, then nodal forces (L , n = 1 to 9) have to be

62
It has been mentioned earlier that an axisymmetrical solution would seem to be the best approximation to analyze the three- Hence, stresses in the bearing system are expected to be larger when using a olane stress or olane strain solution method , [ 12 ] . CHAPTER 5 .

MATERIAL ELElflENT MODELS
The modeling of the bearing system has been discussed in chapter (4), however, the discussion of the material element models which represent the properties 63 and behavior of the different materials used in the construction of such a structure, is left to this chapter.
The different ~aterials used in the construction of the bearing system are divided into three main groups, 1) steel, which is used to manufacture the pot bearings and masonry plates, 2) concrete, which is usually used to build abutments for such structures, and 3) material model to si~ulate the interface between the steel masonry plate and the concrete abutment. It is noteworthy here to point out that there is no such material that exists in nature; however, it is essential for the completion of this investigation to develop certain material models which best approximate the behavior of the interface zone depending on which case of shear transfer is being investigated, see sec. (4.2).
Since it is assumed that the load will be uniformly applied in all loading stages at all times, there were no material elements used to represent the elastomeric material, see chapter (1) Nevertheless, a complete discussion of the elastomeric material properties and behavior is presented in the For axisymmetric analysis the material matrix is, branch. This is described as the elastic -plastic behavior of the concrete, which i.s attributed to the fact that under uniformly applied load, increasinq at a moderate speed, micro cracks within the concrete will take place and become larger as the load increases.
When the internal stresses in the concrete reach the ultimate compressive strength, the concrete will crush.
The yielding stress of the concrete is approximately The slope of the initial straight elastic portion of the stress -strain curve is denoted as the modulus of elasticity (E ) of the concrete, which could be c calculated, with reasonable accuracy, from the empirical eauation E = 33 w 3 / 2 If""' Psi c c ( 5 • 2) where w is the unit weight of the ha.rdened concrete in pcf. Equation ( 5. 2) has been obtained by testing structural concretes with values of w from 90 to 155 pcf.
TPnsile strength (ft) i.s a more variable property than compressive strength, and is about 0.10 f' c to 0.15 f~ [14]. It has been found to be proportional to I F c . ft = 6. 7 The ACI code has indirectly used ~ p si for normal-weight concrete and c = s.7 If' for all light-weight concrete (ACI-11.2).
ft c There are two elastic-plastic material models available in the NONSAP library, elastic-plastic von Mises yield condition, and elastic-plastic Drucker Prager yield condition, see table (3.3). However, the latter is formulated for an elastic -perfectly plastic material behavior; hence the concrete element in this investigation will be represented by the elastic-plastic von Mises yield condition element material model, assuming that it is the best approximation of the concrete stress -strain curve discussed above.

5.2.l von MISES YIELD CONDITION
The von Mises yield criterion considers the octahedral shearing stress as the key variable for 69 causing yield of materials which are pressure independent.
It states that yielding begins when the octahedral shearing stress reaches a critical-value, K, which is defined as the yield stress in pure shear, such that the octahedral shearing stress is, [3]: If the coordinate axis (y, z, x) coincide with the principal directions where s 1 , s 2 , and s 3 are the principal deviatoric stresses, then  It is noteworthy to point out that the stiffness of the springs is increased by selecting a very large (E) , so the magnitudes of the area (A) and the length (L) of the springs will have very little significance.

THE CASE OF INFINITE SHEAR TRANSFER
The finite element grid system representing this case is shown in fig. (4.6) where no special consideration has been given to model the interface between the steel and the concrete. As shown in the grid system, the steel elements are directly connected to the concrete elements through their nodal points. In other words, the interface elements in this case are nothing but the common nodal points of the two different materials.
Hence, not only vertical stresses will be transferred, but shear stresses will be transferred infinitely as well, so that the steel bearing plates and the concrete will behave as a continuum.

THE CASE WITH SHEAR TRANSFERRED THROUGH INTERFACE ELEMENTS
The behavior of the interface between material elements is one of great complexity because of the difficulty in finding the most suitable function that describes the slip. Straight interface elements with linear description of slip between rock joints were first introduced by Goodman, Taylor and Brekke (1968) [15]. The bond behavior between steel and concrete was simulated by Ngo and Scordelis (1967) [16], with the help of linear springs placed between element nodes.
However, this method leads to incompatibilities between nodes if higher order finite elements were used.
Ghaboussi, Wilson and Isenberg ( 19 6 9) [ 17,18] , have also formulated an interface element covering a wide range of joint properties. Schafer (1975) [19], has developed straight bond elements with linear and parabolic behavior of slip. In 1977, Buragohain and Shah [20,21] proposed curved isoparametric line and axisymmetric interface elements with zero thickness to . ulate curved surfaces and parabolic variation of slip. s1m None of the interface elements mentioned above Us ed in this investigation. They were studied in was order to help in developing the interface spring elements used to simulate the behavior of the interface zone between the masonry steel plate and the concrete abutment.
It has been found that there are two different general approaches in modeling the interface zone.
It could be represented by interface spring elements simply transferring horizontal and vertical stresses, or by twodimensional finite elements.
In either approach, the slip could be described by either linear or nonlinear functions as long as compatibility conditions are satisfied.
In the following two sections, two different interface elements, each representing a different approach, will be introduced. The discussion of the interface spring elements used to simulate the interface zone in the bearing system will be left to chapter (6).

82
The formulation of the two-dimensional interface element Ef=vertical stiffness of the IFE at different stress levels.
£c=Normal strain at n whic.h crushing occurs.  i::  fig. ( 6. 4) and fig. ( 6. 5) respectively.                  w plates bending would be higher. Add to that in the case shear transfer the fact that a horizontal spring of zero modulus of elasticity equalling 2 9 Ksi is small but not zero makes the plates bending restraint even higher.
Notice that the effect of using interface spring elements on bending stresses was different in the two-beams example problem presented in sec. (6.3.1).
However, here a different structure is considered, and the interface spring elements connect steel to concrete rather than . steel to steel. The concrete negative vertical and positive shearing stresses were lower in models which included interface spring elements due to the lower bending stresses resulting in those models as discussed above.
The masonry plate was assumed to be rigidly connected to the concrete abutment, therefore, separation of the masonry plate steel elements from the concrete elements is not allowed. Therefore

I-'
Ta bl e ( 7.3 .b) Plan e stra in so luti o n . Ve rt i ca l a nd ho rizontal displac e ment s in top o f th e triangular stee l el e ments adjacent to the inte rface zone .         Pl a ne s t ra i n sol ut ion  ..    Pl a ne strain solutlon. Averaged be nding stresses (Ksi) in the t o p of the mas o n r y steel plate . Gaus s integration points 2 a nd 4.           Results of the study show that: 1) The behavior of stresses and deformation of the bearing system was essentially the same in both the axisymmetr ic and the plane strain solution methods.
However, the numerical difference_ in the results could be attributed to the displacement in the circumferential direction in the axisymmetric elements.
Nevertheless, the axisymmetric solution represents a more realistic approximation of the actual threedimensional solid structure and shows that compression stresses in the concrete are not as high as anticipated in the previous finite element analysis of the bearing system. The stress -strain relation is given by: where €H and s v are the components of the relative strain in the spring connecting points I and J. oH and av are the corresponding stress components. Note that the strain components are shown in the elements local coordinates and they are positive when the spring connecting points I and J is in tension.
Rewriting (A.l) symbolically: (A. 2) where o L and s L are the springs stress and strain components written in local coordinates, and [KL] is the spring's stiffness matrix in local coordinates.
However, if the spring's stiffness matrix is to be added to the structure stiffness matrix, it has to be transferred into global coordinates.
Let the transformation matrix, (A], relate the strains and displacements as such, Note that in this element formulation a linear stress -strain relation was assumed to describe the relationship between bond slip and bond stress, see  The global element stiffness matrix is: (A.19) where [T] is the transformation matrix containing the direction cosines.
The complete global element stiffness matrix for a nondilatant material is shown in table (A.2).  where C and C are defined as in equation (A.17), and nn ss c 88 is added to account for the stress and strain components in the e direction, but its physical meaning is not clear and is assumed to equal zero.
The global stiffness matrix for the axisymmetric element is shown in table (A.4). For more details see the source [17,18].