3DG-C01-00008 (1)

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BECHTEL CORPORATIONENGINEERING

ENGINEERING DESIGN GUIDE FORFINITE ELEMENT MODELING OFCONCRETE STRUCTURES3DG-C01-00008, Rev. 003, October 28, 2002Prepared by: S. Jan/S. WuApproved by: S. Mullen

TABLE OF CONTENTS

LIST OF FIGURES

LIST OF SYMBOLS

1.0 PURPOSE

2.0 FINITE ELEMENT LIBRARY

2.1 Prismatic Beam Elements2.2 Non-Prismatic Beam Elements2.3 Thin Shell/Plate Elements2.4 Solid Finite Elements2.5 Axisymmetric Solid Elements 2.6 Boundary Spring Elements 2.7 Rigid Links

3.0 MECHANICAL PROPERTIES FOR CONCRETE

3.1 Stiffness Parameters3.2 Cracked and Uncracked Sections3.3 Static and Dynamic Properties

4.0 MODELING OF SUPERSTRUCTURES

4.1 General4.2 Finite Element Representation 4.3 Lumped Mass Stick Beam Representation




5.0 MODELING OF MAT FOUNDATIONS

5.1 Finite Element Foundation Model 5.2 Geotechnical Requirements 5.3 Soil-Supported Foundation5.4 Pile-Supported Foundation

6.0 MODELING OF LOADS

6.1 Nodal Loads6.2 Beam Span (Line) Loads6.3 Finite Element Loads6.4 Thermal Loads 6.5 Applied Displacements 6.6 Dynamic Loads

7.0 CHECKLISTS FOR FINITE ELEMENT MODELING

7.1 Checking of Geometry7.2 Checking of Masses and Loads10.1 Checking of Analysis Results

8.0 FINITE ELEMENT CONCRETE DESIGN

8.1 Beam and Column Design8.2 Slab and Wall Design 8.3 Design for Thermal Loads

9.0 MODELING EXAMPLES

9.1 COKER TABLE TOP STRUCTURE 9.2 COMPRESSOR PEDESTAL

10 COMPUTER PROGRAMS

10.1 GT STRUDL (CE029) 10.2 Bechtel SAP (BSAP) Family Programs (CE800)10.3 STAAD III Program

11.0 REFERENCES




LIST OF FIGURES

Figure 1 Finite Element Model for Framed-type Structures(Including Mat Foundation)

Figure 2 Finite Element Model for Shear Wall Structures(Use Thin Shell/Plate Elements)

Figure 3 Finite Element Model for Shear Wall Structures(Use Solid Finite Elements)

Figure 4 Lumped Mass Stick Beam Model(including Soil Springs and Dampers)

Figure 5 Coker Table Top Structure

Figure 6 Concrete Pedestal Supporting Two Units of Compressors




LIST OF SYMBOLS

Ax beam sectional area about local beam x-axis (axial direction)

Ay beam shear areas at local beam y and z-axis, respectivelyAz

Ec modulus of elasticity for concrete

fc’ 28-day concrete compressive strength (psi)

fr tensile strength for concrete

Gs dynamic soil shear modulus

Ie effective cracked moment of inertia of section

Ig gross (uncracked) moment of inertia of section

Icr moment of inertia of cracked transformed section

Ixx area moments of inertia about local beam x, y, and z-axis, respectivelyIyyIzz

ks modulus of subgrade reaction

Mcr cracking moment, fr Ig/y

Ma maximum calculated moment

Mx design bending moments at local element x and y-axis, respectivelyMy

Mxx calculated bending moments at local element x and y-axis, and twisting momentMyyMxy

SFy shape factors at local beam y and z-axis, respectivelySFz




Vs soil shear wave velocity

wc unit concrete weight (pcf)

c Poisson's ratio for concrete

s Poisson's ratio for soil

soil weight density




1.0 PURPOSE

This Design Guide is intended to provide general guidelines for proper modeling for the design and analysis of reinforced concrete structures. The structural systems are considered to be linear and elastic, and may be subjected to static and dynamic loads. Concrete structures which include shear wall and framed-type superstructures on mat foundation are modeled using the finite element method. For dynamic/seismic analysis, techniques for modeling the superstructure as a lumped mass model are also presented.

Winkle type springs are used to model soil media for consideration of the soil-structure interaction effects. The soil media is simulated as frequency-independent soil spring constants (Ref. 11.3). Refer to Civil Design Guide 3DG-C01-00004 (Ref. 11.4) for special consideration of frequency-dependent soil impedance functions for design of nuclear facility structures.

2.0 FINITE ELEMENT LIBRARY

The finite element method is a numerical procedure for solving continuum mechanics problems. It requires simulation of a concrete structure as an idealized finite element computer model with one or more different types of finite elements

GT STRUDL and BSAP (CE800) are the current Bechtel Standard Computer Programs for structural analysis and design. A sample finite element library used in both programs for modeling is presented in the following section. For application of other types of finite elements not discussed herein, refer to User’s manuals of computer programs presented in section 10.0.

2.1 Prismatic Frame Elements

The frame element is a two-node line element which is the most common element constitutive to a finite element model. It is used for modeling beams, girders, columns, and bracing members for resisting axial, bending and torsional forces. The following sectional properties are required as input data to a computer model:

(a) Cross sectional area, Ax

(b) Moments of inertia, Ixx, Iyy and Izz

(c) Shear areas, Ay and Az (or Shape Factors SFx and SFy)

The cross-sectional dimensions such as width and depth or diameter may be specified and lt the program to calculate the above properties.




Rotational degrees of freedom at the end nodes of a frame element can be released to simulate the pinned end condition. Sectional properties , output of forces and moments for beam elements are defined in local element coordinate systems. Refer to Program User's Manuas for definition of beam coordinate systems.

2.2 Non-Prismatic Beam Elements

These elements are very useful for defining variable cross-sectional properties such as haunched columns and tapered beams. The variable depth member is modeled as a series of segments along the member length. Each segment has constant sectional properties Ax, Ay, Az, Ixx, Iyy and Izz. Computer programs will utilize a numerical integration scheme to calculate each individual element stiffness.

2.3 Thin Shell/Plate Elements

Thin shell/plate elements are generally defined as quadrilateral (4 nodes) or triangular (3 nodes) finite plate elements. These elements are commonly used for modeling walls, slabs, and mats. They exhibit out-of-plane bending and in-plane membrane stiffnesses. In-plane rotational stiffness is not considered in the programs. Therefore, the user should provide either a restraint or a very soft stiffness for the in-plane rotational degree of freedom of a plate/shell element. However, a small in-plane rotational stiffness is provided by GTSTRUDL/BSAP and additional rigidity may not be required.

The effect of transverse shear deformation is not considered in GTSTRUDL and BSAP programs for thin plate/shell elements. Therefore, the element stiffness may be too rigid for a moderate thick wall, floor or mat. In this case, layered solid (BRICK) elements or THICK PLATE elements should be used.

The element mesh sizes should be properly selected so that the aspect ratio is not more than three, and the internal angle is not less than 30 degrees.

Output of element bending moments and membrane forces are used to size main reinforcement of a concrete section. These forces are shown in the computer output. Transverse shear forces are used to check concrete shear stresses and to calculate the required shear reinforcement.

The following table provides a sample listing of thin plate/shell elements available in GTSTRUDL and BSAP computer programs:




Program Element type Number of Nodes &Shapes

Transverse ShearForces

GT STRUDL SBHQ6 4-Quad YesIPBQQ 8-Quad YesSBHT6 3-Tri Yes

BSAP* PLATE 4-Quad No3-Tri No

LCCT9 4-Quad No3-Tri No

GSHELL 4-Quad Yes3-Tri Yes

* Note: A new plate element with transverse shear force output has been incorporated in BSAP Windows version which will be released in 2003.

2.4 Solid Finite Elements

Solid finite elements have been extensively used in modeling of facility structures such as the containment structure, internal structure and mat foundation. They are used to account for the shear deformation effects and non-uniform distribution of thermal gradient across walls and non-linear effects of thick walls and thick mat. Several layers of solid elements should be used across the thickness of walls or mat to represent the stress distribution. Outputs of solid elements represent the stresses at element centroids or at the centers of plane element faces. Stresses at element nodal points can also be printed out. The section forces and moments for reinforcement design are computed by programs by performing stress integration across the thickness.

The following table provides a sample list of the solid elements available in GT STRUDL and BSAP computer programs:




Program Element Type Number of Nodes &Shape

Post-processor for Section Force Computations

GT STRUDL IPLS / IPSL 8-Hex YesTRIP 6-WEDGE Yes

BSAP (CE800 ) BRICK 8-Hex Yes (BSAP-POST)

2.5 Axisymmetric Solid Elements

Axisymmetric solid elements which consist of 4-nodes and 3-nodes solid elements, are used for modeling axisymmetric shell-of-revolution structures, such as a containment structure, or circular containment mat foundation under axisymmetric or anti-symmetric loads. When the loading is not symmetric, a Fourier series method can be used to decompose the loading as the sum of several symmetric component loadings. The analysis is then performed for each component, and the solutions are the summation of each component result.

Outputs of element stresses are defined at the centroid of elements in the radial, meridional, and hoop directions. A post-processor program may be required to calculate section forces and moments for reinforcement design.

The following table provides a listing of axisymmetric solid elements available in Bechtel Standard Computer Programs:

Program Element Descriptions

GT STRUDL IPQL 4-Nodes solidIPQLQ1 5-Nodes solid

BSAP RING 4-/3-Nodes solid

2.6 Boundary Spring Elements

Boundary spring elements are the basic element type constitutive to a finite element model. It is used to simulate the flexibility of foundation supports or boundary support condition (nodal restraints) in three translations and three rotations. A soil-supported or pile-supported foundation can be analyzed by simulating soil or pile stiffnesses by sets of boundary spring elements attached to nodal points of the foundation model. An infinitely




rigid support condition can also be modeled by boundary spring elements with very high stiffness. Outputs for the spring elements are reaction forces and moments with respect to their axes. These outputs can then be used for soil pressure and settlement calculation or anchorage support design.

In GT STRUDL, a nonlinear spring element capable of compression-only behavior is available for modeling foundation under a significant up-lift condition. It should be noted that the structural analysis becomes nonlinear analysis when soil is considered to exhibit compression-only behavior, A linear load combination of individual load analysis results is not valid. The FORM LOAD command in GT STRUDL can be used to form an individual load case from several basic load cases. The nonlinear analysis is then carried out based on the formed individual load cases.

2.7 Rigid Links

In finite element modeling, structural members are normally inter-connected at member centerlines. The analysis results will cause an overly conservative design for a concrete structure with large column dimensions, deep beams or thick floor slab and foundation. The rigid links may be used at the member connection points. It may.also be used to simulate the connection between heavy equipment and the supporting concrete structure.

3.0 MECHANICAL PROPERTIES FOR CONCRETE

3.1 Stiffness Parameters

The finite element analysis of concrete structures requires inputs of the following parameters. The following parameters are the basic element input data for calculation of concrete element stiffnesses:

(a) Modulus of elasticity (Ec)

(b) Poisson's ratio (c)

(c) Section areas and area moments of inertia

3.1.1 Modulus of Elasticity (Ec)

Tests have shown that for normal concrete strengths, the modulus of elasticity of concrete (Ec) is found to be a function of the compressive strength, fc’, and unit concrete weight, wc, and can be represented as follows (see Ref. 11.2 and Section 8.5 of Ref. 11.1):




Ec = 33 wc3 fc'

For normal weight concrete, Ec can be obtained as follows:

Ec = 57,000 fc'

Where Ec and fc’ have the units of pounds per square inches (psi), and wc is pounds per cubic feet (pcf).

Since plain concrete exhibits limited tensile capacity (see Section 3.2), Ec determined from the above equations was derived from compression tests. For practical analysis purposes, both the tension and compression modulus of elasticities are assumed to be identical. Also, based on the ACI report (Ref. 11.7), the above ACI expressions may tend to overestimate the modulus of elasticity for high strength concretes with fc’ in excess of 6000 psi. In this case, approximate formulas presented in Ref. 11.7 may be used, or tests may be required to determine the Ec based on the actual design mixture.

Note that concrete compressive strength and the modulus of elasticity increase with time. At the end of 1 year, the value of compressive strength may be 1.25 times that of the value for the 28-day concrete. Also, in accordance with Ref. 11.8, the concrete modulus of elasticity may be increased by about fifty percent. Unless dictated by Client's requirements or Chief's approval on certain special conditions, fc’ used for design and analysis of concrete structures should be the compressive strength that corresponds to 28 days of age. Higher concrete strengths may be used only for special situations or for reconciliation of as-built conditions.

3.1.2 Poisson's Ratio (c)

Poisson's ratio of concrete is required in the finite element analysis to account for the shear deformation of beam elements or to calculate the membrane and flexural stiffnesses of plate elements. Poisson's ratio of 0.17 should be used in the finite element analysis of concrete structures regardless of concrete strength, age, loading rate, and moisture content.




3.1.3 Section Areas and Area Moments of Inertia

For calculating axial and bending stiffnesses of beam elements, the program requires input of areas (Ax, Ay and Az) and moments of inertia (Ix, Iy, and Iz). Areas and moments of inertia of a concrete beam section are calculated based on the basic principle of mechanics. For stiffness calculation of plate and shell elements representing slabs, walls and mats, the thickness of plate elements is required.

Area moments of inertia can be calculated either based on the gross (uncracked) section or the effective cracked section. Discussions of the concrete cracked behavior are presented in Section 3.2.

3.2 Cracked and Uncracked Sections

It has been found that the tensile strength (fr) of a concrete member in flexure is about 10 to 15% of the compressive strength. For normal weight concrete, (fr) can be expressed as follows (see Ref. 11.2 and Section 9.5.2.3 of Ref. 11.1):

fr = 7.5 f c'

Where the stress level is in excess of fr, cracking of concrete sections becomes imminent. In this case, the effect of concrete cracking on the modeling of concrete structures should be considered, especially for cases where the deflection criteria are governing the design. Iteration of the finite element analysis may be required by using the effective cracked sections obtained based on the following ACI recommendation (Section 9.5.2.3 of Ref. 11.1):

Ie = (Mcr/Ma)3 Ig + (1 - (Mcr/Ma)3) Icr

where Ie = Effective cracked moment of inertia of section

Mcr = cracking moment, fr Ig/yt

Ma = maximum moment

Ig = gross (uncracked) moment of inertia of section

Icr = moment of inertia of cracked transformed section




yt =distance from centroidal axis of gross section, neglecting reinforcement, to extreme fiber in tension

For massive concrete structures where calculated stresses are expected to be low, the modeling of concrete sections can be based on the gross (uncracked) section. Also, the gross concrete sections may be used with reasonable accuracy in the static analysis when the strength criteria are governing the design, due to the fact that distribution of overall applied loads to individual members is based on the relative stiffnesses of concrete sections.

In the dynamic analysis of concrete structures, calculation of the dynamic response depends primarily on the resonant frequency parameter, defined as the forcing frequency over the structural natural frequencies. Since cracking can reduce the flexural stiffness of concrete structures, and thus reduce structural natural frequencies, justification should be provided if the gross concrete sections would be used in the dynamic analysis. Otherwise, iteration of the finite element analysis may be required to account for the concrete cracking effect.

Note that gross (uncracked) concrete sections have been used in the seismic analyses of heavy massive nuclear concrete structures, without resorting to a time-consuming iteration process in accounting for the concrete cracking effect. This is due to soil-structure interaction effect on seismic response. Also, the input seismic motions are represented as broad band spectra. Therefore, any changes in the concrete stiffnesses due to concrete cracking would have insignificant effect on overall structural seismic response.

3.3 Static and Dynamic Properties

The formulas presented in Section 3.1.1 for evaluating the concrete properties were derived based on the static test condition. It has been shown from dynamic tests (Ref. 11.5) that both dynamic modulus of elasticity and concrete strength increases with increasing dynamic strain rates, and thus is higher than the static value. In Appendix C of Ref. 11.6, it is suggested that the concrete strength be increased by a factor of 1.25 for axial and flexural compression members subjected to dynamic transient impulsive and impact loads.

However, unless dictated either by Client requirements or through chief's approval for special conditions, the concrete properties defined in Section 3.1.1 that correspond to static condition should be used in development of the dynamic concrete model.

It should be noted that dynamic soil spring stiffness is significantly higher than the static value.




4.0 MODELING OF SUPERSTRUCTURES

4.1 General

Design of complex concrete structures generally requires the use of a finite element computer analysis for determining stress distribution to concrete members. For performing a computer analysis, accurate finite element modeling of the superstructure is the first and most important step to be undertaken to assure the accuracy of the analysis results. Modeling requires an understanding of the structural behavior and level of accuracy needed and involves the subdivision of the structure into an assemblage of appropriate discrete finite elements, interconnected at a finite number of joints or nodal points. This analytical idealization for developing a three dimensional finite element computer model can be accomplished by utilizing the finite element library presented in Section 2.0. Analyses based on a three dimensional finite element computer model are generally preferable, since most of the analysis programs have been integrated with the design programs as a complete automated analysis and design program package, and the hardware capacity has been significantly increased.

In the dynamic/seismic analysis of a complex and heavy structure, a simplified finite element model or lumped mass stick beam model may be used to evaluate the overall structural dynamic responses such as shears, moments, displacements and acceleration in-structure response spectra. For more accurate analysis and design, a detailed finite element model should be used.

Discussions of the finite element and lumped mass stick (see Figure 4) representation for a concrete structure are presented in the following sections.

4.2 Finite Element Representation

Two commonly constructed industrial structures are framed-type and shear wall structures, or a combination of both types. Finite element representation for these two types is presented as follows:

4.2.1 Framed-Type Structure

Framed-type structures are moment-resisting frames, which typically consist of beams, columns, and floor slabs (see Fig. 1). A symmetrical and regular structural arrangement such as pipe rack bents can be modeled as a two dimensional (2-D) model in order to reduce the time-consuming 3-D modeling effort. Numerical solutions for the 2-D model can be obtained very efficiently due to the reduction of the number of degrees of freedom from six to three (two translations and one rotation) per nodal joint. However, since loading patterns are generally very complex and structural arrangements are generally




not symmetric, use of a three dimensional finite element model is always preferable. Modeling of beams, columns and floor slabs are presented as follows:

4.2.1.1 Beams and Columns

Beams and columns are subjected to axial, bending and torsional forces. They are modeled as two-node prismatic or non-prismatic frame elements. Rigid links should be used at beam-column intersection points when floor beams are not located at the same elevations, and/or are not intersecting with the same nodes of columns. The stiffness properties should be defined as discussed in Section 3.0. For design purposes, most of the computer programs do have the capabilities of printing out the maximum moments and forces at various locations of a frame element.

In the dynamic model, an additional nodal point should be defined as a minimum at the center of beams in order to simulate the first vibrational mode characteristics. For analysis of a large turbine-generator foundation, more additional nodes along a member may be required in order to reach operating frequency ( Reference 11.3 ).

4.2.1.2 Floor Slabs

Floor slabs subject to vertical loads should be modeled as finite thin shell/plate elements for consideration of the slab flexibility in two directions. For horizontal loading cases, floor slabs may be considered to be infinitely rigid by using rigid links to connect the nodes between columns. Rigid links can also be used when the simulated plate elements are not intersected at the same elevation due to different thicknesses of slabs. Accuracy of finite element results depends on the level of mesh refinement of a model and the aspect ratio of an element. The aspect ratio of quadrilateral plate elements between two adjacent sides should be not more than three (2 is the most desirable). For triangular plate elements, the angle between any two adjacent sides should be larger than 30 degrees. See Section 7.0 for detailed discussions of checklists for finite element modeling.

The finite element model developed above can be used in the dynamic/seismic analysis for evaluation of vertical floor response spectra in order to account for floor flexibility effect in vertical direction. The calculated vertical spectra should be used for seismic qualification of nuclear power plant structures, systems and components.

4.2.2 Shear Wall Structures

Shear wall structures are designed to resist vertical and lateral loads. Such structures are modeled as an assemblage of finite thin shell/plate elements (see Fig. 2) or finite solid elements to simulate the thick wall effect (see Fig. 3). The model should consist of a sufficient number of plate elements with bending and shear actions to obtain accurate




results. The limiting aspect ratio and the internal angle of plate elements should follow the same criteria as discussed in Section 4.2.1.2. When the solid finite elements are used to simulate the thick wall effect, a minimum of three layers should be modeled so that section forces and moments can be adequately evaluated from the element stress resultant output. Refer to Section 7.0 for checklists for finite element modeling.

In the dynamic/seismic analysis, shear walls are generally considered to be massless and are modeled to represent only the stiffnesses of a structure. Masses are lumped at the mass center of floor slab elevations. This type of modeling greatly reduce the dynamic degrees of freedom of a structural model and an adequate solution can be achieved for obtaining the dynamic responses.

4.3 Lumped Mass Stick Beam Representation

The lumped mass stick beam model (see Fig. 4) has been commonly used in seismic analysis of nuclear power plant structures for gross estimation of structural responses, such as shears, moments, displacements and acceleration in-structure response spectra. This approach is a cost effective way for evaluating structural dynamic characteristics, particularly when a detailed 3-D soil model needs to be included in the soil-structure interaction seismic analysis. This dynamic model requires the calculation of the mass centers and rigidity centers, and beam element properties. Total tributary masses in six dynamic degrees of freedom (three translations and three rotations) calculated between two floor slabs are lumped at the mass centers of floor slabs. The stiffnesses of shear walls or columns are represented as frame elements, and are located at the rigidity centers. Rigid links are then used to connect the mass centers and the rigidity centers. Note that calculation of the mass center and the rigidity center is not required when the finite element representation is considered in the modeling, since they are automatically accounted for in the finite element analysis.

5.0 MODELING OF MAT FOUNDATIONS

5.1 Finite Element Foundation Model

A mat foundation is designed to support a superstructure transferring loads from columns, walls or piers to soil media or piles underneath the foundation in order to avoid excessive settlement and to provide stability. A computer analysis is generally required for design of a mat foundation due to the complexity of loading patterns applied to the foundation, and also to account for the flexibility of the mat foundation. For preliminary design purposes or for design of small sizes of spread footing, a design based on the rigid foundation concept may be employed.

In the finite element computer modeling of a mat foundation, the foundation is simulated as an assemblage of plate/shell elements, and is discretized into a number of




quadrilateral or triangular finite plate/shell elements (see Fig. 1). Loads from superstructure may be applied as concentrated forces at nodes or uniform pressure on elements. For framed concrete structures, the rigidity of the superstructure may be neglected. For thick shear wall structures, the effect of wall stiffnesses at the wall-mat junctions may require consideration. Note that the effect of the structural rigidity is automatically accounted for, when a complete three dimensional finite element model including the superstructure is utilized in the analysis.

For a mat foundation considered to be sufficiently thick to invalidate the assumption of modeling the foundation as thin plate elements, the multi-layer solid finite elements with eight-nodes should be used. Both GT STRUDL and BSAP have capabilities to perform stress integration through the thickness to obtain stress resultants of finite elements for concrete reinforcement design.

Note that computer outputs of element stress resultants should reflect the same reference or global coordinate systems, since difficulty may result in reviewing the analysis results or sizing the main reinforcement if stress resultants are defined in different element local coordinate systems. The PLANAR COORDINATE in GT STRUDL and the REFERENCE PLANE in BSAP are used to define the common coordinate system for element stress resultant output.

5.2 Geotechnical Requirements

To develop a finite element foundation model, soil or pile-supported media should be simulated as a set of soil or pile spring constants (three translations and three rotations). These spring constants can be obtained from Project Geotechnical Engineer/Consultant or can be determined based on the procedures presented in Ref. 11.3. The geotechnical data required for analytical evaluation of the soil or pile spring constants and for foundation design are:

(a) Soil weight density ();

(b) Poisson's ratio (s );

(c) Modulus of subgrade reaction (ks) for static analysis, or dynamic soil shear modulus (Gs) or shear wave velocity (Vs) for dynamic snalysis;

(d) Allowable soil pressure or allowable pile loads (lateral and vertical);

(e) Soil/pile spring constants and damping values.

For development of the foundation soil/pile spring model, it is suggested that a range of soil properties, such as modulus of subgrade reaction or shear modulus (upper, average




and lower bound values), should be considered in order to account for the variation of the above soil properties. Also, since the analysis results are sensitive to the soil/pile spring model used in the analysis, use of the parametric studies would yield an adequate foundation design. For strength design, a reduced static soil stiffness will yield higher bending moments in the mat, therefore resulting in a conservative design.

5.3 Soil-Supported Foundation

For a mat foundation supported on soil, Winkle type springs are used to simulate soil foundation media to account for the soil-foundation interaction effect. Soil springs attached at nodes are calculated using the modulus of subgrade reaction multiplied by the tributary areas of those nodes. Alternatively, procedures developed based on the elastic half-space theory (Ref. 11.3) can be used to calculate the total translational and rotational soil stiffnesses for a given foundation size. Distribution of the total soil stiffnesses to the nodes of elements is also based on the tributary area of those nodes. Since soil is considered to resist compressive force only, iterations of the finite element analysis may be required to remove vertical soil springs that are under tension. This should be applied to those load combinations that result in foundation uplifting cases.

Soil reactions obtained from a finite element foundation analysis can be used to calculate the maximum soil contact pressure and compare against the allowable. Outputs of moments and shears for finite plate elements are used to size foundation reinforcement (See Section 8).

In the region of high seismic zone and/or high wind area, the foundation may have significant uplift. In this case, the compression only spring should be used. GT STRUDL has the capability of performing the iteration process automatically so that soil reactions are all under compression. It should be pointed out, the result from a linear combination of a set of non-linear analysis results is invalid. All loads such as dead load, seismic, and live loads, etc. must be combined as an independent load prior to the non-linear analysis using the FORM LOAD command.

5.4 Pile-Supported Foundation

Due to poor soil condition, a mat foundation may be designed to be supported on piles. Piles are simulated as translational and rotational pile springs attached at each pile location. The rotational stiffness is normally neglected because of the minimumal effects on the analysis. Pile spring constants can be obtained from Project Geotechnical Consultant based on hinged or fixed pile head condition. Alternatively, procedures developed based on Novak's formulas (Ref. 11.3) can be used to calculate pile stiffnesses. Since piles can resist both tension and compression, iterations of the finite element analysis may be required only when the maximum pile load exceeds the allowables.




6.0 MODELING OF LOADS

Loads applied to a concrete structure are idealized as nodal or finite element loads. The applied loads are then combined with appropriate load factors in accordance with the ACI code or Project Design Criteria for design of concrete structures. See Bechtel design guide (Ref. 11.10) for standard loads and load combinations Five basic load types including dynamic loads are discussed herein. For modeling of other special load types, refer to computer program capabilities described in User's Manuals.

6.1 Nodal Loads

Nodal loads are concentrated loads applied at nodal points (element intersection points). Equipment weight, pipe support loads, wind and/or seismic loads can be simulated as nodal loads.

Nodal loads are defined in global coordinate systems.

6.2 Beam Span (Line) Loads

Beam span loads are element loads applied to beams or columns as concentrated forces, weight density or distributed loads with uniform, partial uniform, triangular, or trapezoidal shapes. Beam span loads are generally defined in local element coordinate systems.

Weight density or accelerations can also be applied to beam elements for calculating element self weights or uniform applied seismic loads. Directions of load application should be defined in global coordinate systems.

6.3 Finite Element Loads

Finite element loads are defined as loads applied uniformly to finite plate/shell elements. These loads may be simulated as weight density, accelerations in any directions, surface pressure normal to elements, or edge loads along the sides of elements.

6.4 Thermal Loads

Thermal loads are normally specified as uniform temperature differences along beam elements and/or as temperature differences across the beam depth. However, they are applied as nodal temperature difference and or temperature gradient for plate/shell type elements.




The thermal stress is very sensitive to mesh density, temperature distributions, and boundary restraints. A coarse mesh size will result an artificially stiff system and therefore results in a larger thermal stresses. Temperature distribution should be as uniform as possible. Steep change in temperature profile will cause unreasonable large thermal stresses in the adjacent elements. The stiffer the restraint, the larger thermal forces will be. Therefore, it is important to properly define the boundary restraints such as member end releases, soil/pile springs and support joint releases.

6.5 Applied Displacements

Applied displacements are used to simulate the effects induced by support settlements or differential movements between structures. They are defined at nodes, and are generally defined in terms of global coordinate systems.

6.6 Dynamic Loads

Dynamic loads are generally defined as sinusoidal forcing functions or time history functions digitized as forces or accelerations versus time. These loads are applied to a finite element model as a nodal forcing function or ground excitation forces. They are used to calculate the dynamic structural response based on applicable dynamic analysis methods. Dynamic loads such as steady-state sinusoidal forcing functions are induced typically from imbalance of vibrating machines. Seismic forces are simulated as acceleration or displacement time history input motions or response spectral data, and applied at the foundation base of a model. The other dynamically induced loads can be defined at any nodal point of a model as nodal forcing functions.

7.0 CHECKLISTS FOR FINITE ELEMENT MODELING

Having developed a finite element computer model to simulate a concrete structure, a pre-processor (data check run) should be performed first to check the geometry of the model to ensure that nodal points and finite elements are properly connected, and loads are properly applied at appropriate nodes or elements. Any error messages appearing on the output should be carefully checked and resolved. The following checklists are to provide guidelines for checking computer models, masses/loads, and analysis results:




7.1 Checking of Geometry

(a) Units

Units to define member/element lengths, element thicknesses, material properties, section properties, and coordinates must be checked to ensure consistent units or default units prescribed by programs are used.

(b) Geometry Plot

Geometry of computer models should be plotted and checked to assure the accuracy of overall structural geometrical modeling. The finite element model can be plotted to check whether elements are missing or overlapping each other. The elements so plotted are shrunk as "shrink shapes" so that all element edges become visible.

Note that since most of the computer programs have the capability of automatic mesh generation for developing the geometry and finite element meshes, these options shall always be used so that the input coding errors can be minimized.

(c) Element check

In order to obtain accurate finite element analysis results, the user should be aware of the limitations of the element aspect ratio, internal angular and warping of a mesh size. General guidelines for developing appropriate finite element mesh sizes are discussed as follows:

i) Aspect ratio

In a finite element model, the aspect ratio is defined as the ratio of the longest to the shortest element dimensions. The ratio should be maintained to be smaller than 3 (2 is ideal) for a quadrilateral element and 1.5 for a triangular element.

ii) Skewness/Distortion

Finite elements with a near rectangular shape should be used to obtain the best numerical solutions. In other words, the corner angle of a quadrilateral or solid elements should be close to 90 degrees. If the angles of a quadrilateral element become acute, the computational accuracy will be lost. This is also true for non-parallel solid elements. Therefore, it is suggested to develop the meshes of finite elements in such a way that their angles are never less than 30 degrees or greater than 150 degrees.




iii) Warping

Warping of a finite element occurs when four nodes of any quadrilateral elements are not in the same plane. The stiffness matrix of a warped element would yield abnormal or even erroneous results. The maximum warping angle (defined as an angle between the two normal vectors which are perpendicular to the planes that pass through two subdivided three-node quadrilateral elements), as permitted by computer programs, should be less than 10 degrees.

Note that typically the above element checking procedures are provided in the programs. Each program has its own definitions of quantitative measure of these parameters.

(d) Support conditions

Support boundary conditions whether simulating as rigid or flexible supports should be properly modeled. Support can be checked using computer display or from the computer printout. Special attention should be given where the stiffness of finite spring supports or values of nodal displacements are specified.

(e) Structural Instability

Structural instabilities are normally caused by improper member end release, insufficient boundary restrains, and improper member connections. It is very difficult to locate the problem since most of the analysis programs can not identify the exact nodal point which has the singularity during equation solving. However, the PERFORM INSTABILITY command in GT Strudl can be used to locate the actual area of the structure having the instability problems.

7.2 Checking of Masses And Loads

The computer output of total masses (or weights) and total applied loads for each loading condition should be checked against the hand-calculated values. Any discrepancies should be identified, and resolved.

7.3 Checking of Analysis Results

Analysis results of nodal displacements, mode shapes and/or stress resultant contours should be plotted by post-processor programs to assure the accuracy of structural behavior. Total reactions at boundary support points should equal the summation of the applied loads for each loading conditions.




In dynamic analysis, it is necessary to consider a sufficient number of vibration modes to assure participation of all significant modes so that at least 90% of the total mass is included. The inertial forces due to the missing mass in dynamic analysis should be recovered for final structural design. The FORM MISSING MASS command in GT STRUDL can be used to calculate the missing inertial forces.

8.0 REINFORCED CONCRETE DESIGN

8.1 Beam and Column Design

GTSTRUDL can perform reinforced concrete design for beams and columns based on ACI 318 and BSI (CP110-72 and BS8110) codes. It is recommended that at least three sections, at member ends and center, need to be checked against the Code. In addition, points of large concentrated loads need to be checked. If p-delta effect is not considered in the analysis, a proper moment magnification factor should be specified for each column.

The special code requirements for high seismic zones are considered in GT STRUDL for sizing and detailing the reinforcement.

8.2 Slab and wall design

The outputs of PLATE/SHELL elements from a finite element analysis include the bending moments Mxx and Myy, twisting moment Mxy, transverse shears Vxx and Vyy, in-plane shear Vxy and axial forces Nxx and Nyy. For design purpose, an average resultant over a section width of two times of element thickness plus column width (or, concentrated load area) may be used. Contour plots of stress resultants may also be used to evaluate the design forces and moments The CALCULATE SECTION COMMAND in GT STRUDL can be used to obtain a total cross-sectional stress resultants.

To account for the twisting effect, the design moments Mx and My can be conservatively calculated as follows:

Mx = ABS(Mxx) + ABS(Mxy)

My = ABS(Myy) + ABS(Mxy)

The signs of the design moments Mx and My should be the same as those for Mxx and Myy, respectively. It is noted that Mxx (or Myy) and Mxy must be the moments in an element under the same load condition.




Alternatively, the angle sweep method in the OPTCON module of the BSAP-POST program may be used. OPTCON calculates the design moments based on the following equation for angles from zero to 180 degrees with increments of 10 degrees (nineteen sections):

M = (Cos2 )Mxx + (Sin2 )Myy + 2(Cos )(Sin )Mxy

The calculated moments are used to design the reinforcement for each of the nineteen sections. An optimization technique is then used to obtain the required reinforcement.

8.3 Design for Thermal Loads

Forces and moments due to thermal loads are self-limited in the nature. Concrete will crack when the concrete stress reaches the concrete tensile strength of fr. Consequently, thermal moments and forces will be released.

ACI 349 Appendix A provides three approaches to handle the thermal stress problems:

a) Structural analysis treats thermal loads acting on a monolithic section and evaluates the rigidity of the section and stiffness of the element based on full uncracked cross section. This approach may be overly conservative because it does not consider the self-relieving nature of thermal stress due to cracking and deformation.

b) Structural analysis considering the cracking of concrete for all loads, mechanical and thermal. This approach is the most accurate and generally results in the largest degree of self-relieving thermal stress, it is very complex, involving significant nonlinear analysis and iterative processes.

This approach may be simplified by performing a finite element analysis based on un-cracked section for both thermal and mechanical loads first, and then for each loading combination, all elements with tensile stresses greater than concrete tensile strength fr( 7.5 f c

' ) are considered cracked. The thickness of the cracked elements are then modified based on the effective cracked moment of inertia. For convenience, the Young’s modulus of elasticity of the elements can be modified to obtain an equivalent cracked moment of inertia of the section. Stress contour plots can be used to identify the area of high tensile stresses. It is still an iterative approach that is time comsuming. However, the number of iterations may be minimized by engineering judgement.

c) Structural analysis considering the structure uncracked for mechanical loads and only considering the effect of cracking on thermal loads. This approach can be accomplished by using OPTCON module of CE800 post processor program (CE217).




The first step of the design approach is to perform a finite element analysis based on uncracked section for both thermal and mechanical loads. The thermal moments obtained from the analysis are then reduced based on the concrete cracked section by using OPTCON. Since the bending moment due to thermal differential is dependent on existing state of stress, OPTCON determines the stress and strain distribution under each loading combination of mechanical and thermal loads. A direct solution to this problem is not possible since both concrete and reinforcement stress-strain relationships are nonlinear. Therefore, OPTCON uses a numerical solution to modify the thermal moment and axial forces based concrete cracked depth and considering the yielding of reinforcement. This approach is simple and recommended.

Design guide 3DG-C01-00002 (Ref. 11.9) provides methods OR calculating thermal stresses.

9.0 Modeling Examples

Modeling of typical reinforced concrete structures are presented in the following sections. These structures have been successfully analyzed and designed for static and dynamic loads by using GT STRUDL.

9.1 Coker Table Top Structure

In Delayed Coker Project, the most heavy and important structure is the concrete table top supporting heavy coke drums. In general, a concrete table top is designed to support two coke drums. A coke drum is normally has a diameter of 25 feet and 110 feet in height. The operating weight of a coke drum is about 6000 kips.

A table top structure consists of a 6 ft thick foundation mat, six columns (8 ft by 8 ft) in a two-by-three arrangement, and a 9 ft thick operating deck. One side of the structure may have a structural wall extending from the foundation mat to near the mid-height of the columns. There are deep girders (8 ft by 8 ft) connecting the columns, also located at near mid-height. The operating deck has large openings to accommodate the coke drum. The height of the table top structure is about 100 feet.

The table top structure is modeled as a 3-D finite element model as shown in Figure 5. The foundation mat, structural wall, and the operating deck are modeled by plate elements. The columns and girders are modeled as frame elements. The soil springs beneath the foundation mat are also included. Due to the large dimensions in columns, girders and the operating deck, rigid links should be used at connection joints of these structural elements. This is essential for obtaining realistic design forces.

The computer model may also include the drums as frame elements. All drum loads including dead weight, seismic and wind loads can then be directly applied to the drum




frame elements. Due to the thickness of operating deck (9 ft) and the heavy drum loads, rigid links between the coke drum (as line element) and the supporting deck ( as finite elements) may be used to maintain plane surface around the opening. This condition can be represented by translational rigid links. The rotational rigid link about the radial axis of bolt circle may be used if the drum base is rigid at the deck connection. The rotational rigid link about the axis tangential to the bolt circle should not be used. This is because the magnitude of this rotational restraint is minimal and difficult to calculate. The restraint of rotational degree of freedom about the vertical axis is inconsequential to the analysis.

If the coke drums are not included in the model, all the drum loads are applied as nodal forces around the bolt circle. Vertical loads are assumed to be uniformly distributed to nodes at bolt circle. Horizontal seismic and wind loads are applied as couples of vertical nodal forces. The following equation may be used to evaluate the nodal couple forces:

F = M/(R)*COS()

This approach of applying nodal forces will result in a conservative design.

The critical design moment at the operating deck is normally at the center column strip between the two drums due to the static vertical loads. For high seismic or strong wind loads, the critical bending moments may occur at the edge column strips.

9.2 Compressor Pedestals

The horsepower of compressors have substantially increased in the past decade. Subsequently, the supporting concrete pedestals have increased in size. In industrial plant such as petro-chemical, LNG and power plant, it generally requires several units of compressors to be supported on a common foundation due to the limitation of plant area. The effect of interaction between units on dynamic responses during machine operation becomes significant. In addition, a dynamic response spectra analysis may need to be performed to evaluate seismic forces for a plant located in high seismic zone. Therefore, the design of a compressor pedestal becomes the most import and complex task in the project design works.

Figure 6 shows a typical 3-D finite element model of a concrete pedestal supporting two units of heavy compressors. The structure consists of a 3 ft deck, 4 ft mat, and 3 ft by 3 ft columns. Deck and mat foundation are modeled as plate elements. The mat is supported on soil which is represented as boundary spring elements.

The following design steps are normally taken in the course of designing a compressor pedestal:




1. Acquire machine data & loading;2. Determine seismic loading or design spectra (if applicable);3. Geotechnical data;4. Other design loads such as dead and live loads;5. Design criteria;6. Sizing pedestals and foundation;7. Finite element modeling;8. Dynamic analyses to assess vibration responses;9. Seismic response spectra analysis (if applicable);10. Static analysis (with UBC earthquake loads if applicable);11. Determine member design forces (in high seismic zone, evaluating seismic design

forces based on steps 9 and 10);12. Reinforced concrete design using applicable code.

The model shown in Figure 6 can be used for both static and dynamic analyses. Bechtel design guide 3DG-C13-00001(reference 11.3) provides detailed guidelines to design a turbine pedestal.

10.0 COMPUTER PROGRAMS

GT STRUDL and BSAP (Bechtel Structural Analysis Program – CE800) are Bechtel Standard Computer Programs. They are validated and documented in accordance with various QA code requirements. Therefore, they can be used for nuclear and non-nuclear applications.

STADD III has been widely used in the industry. Both GT STRUDL and BSAP programs provide interface capability to read STADD III input data. STADD III does not have a QA procedure for program validation and error notification. Therefore, it can not be adopted by Bechtel as a Bechtel Standard Computer Program. Projects use this program has to perform program verification and documentation.

10.1 GT STRUDL ( CE029)

GT STRUDL is a Bechtel Standard Application Program developed by George Tech Research corporation. It is a computer-aided structural engineering software system for assisting engineers in the structural analysis and design process. GT STRUDL completely integrates graphical modeling, frame and finite static, dynamic, and nonlinear analysis, finite element analysis, structural frame design, graphical result display, and structural database management into a powerful menu-driven information processing system.




GT STRUDL is validated and certified in full conformance to the applicable provisions of the United States Nuclear Regulatory Commission software quality assurance andquality control regulations. It performs general purpose structural analysis and design, as well as structural database processing, on a very broad range of structural problems.

10.2 Bechtel SAP (BSAP) Family Programs:

BSAP family programs consist of BSAP (CE-800), BSAP-Pre (CE-212), BSAP-Post (CE-217) and BSAP-Dynam (CE-207). These programs are used for performing linear, elastic analysis and design of any three-dimensional finite element structural models subjected to any static and dynamic loadings including the steady-state dynamic load. Seismic analysis methods based on response spectral and time history techniques are also available for design and qualification of nuclear power plant structures and components.

The OPTCON module of the BSAP-POST program has been commonly used for design of reinforced concrete structures based on the ACI-318 and ASME Section III Division 2 Codes. The effects of concrete cracking and yielding of reinforcing steel on thermal moment calculation are considered in the program.

The above programs are Bechtel Standard Application Programs. They are operating on VAX system, and DEC 5000, and IBM RISC 6000 workstations. The windows based version of the program will be available in 2003.

10.3 STAAD III Program

STAAD III is a general purpose finite element computer program. It is used for performing three-dimensional finite element analysis and design of steel and concrete structures. A dynamic option for calculating structural natural frequencies is also available.The program is operational on the PC computer with Windows operating system.

11.0 REFERENCES

11.1 ACI 318-02, Building Code Requirements for Reinforced Concrete and Commentary.

11.2 C. K. Wang and C. G. Salmon, Reinforced Concrete Design, 4th edition.

11.3 Civil Design Guide 3DG-C13-00001, "Concrete Framed Structures Supporting Vibrating Equipment."




11.4 Civil Design Guide 3DG-C01-00004, "Seismic Analysis of Structures and Equipment for Nuclear Power Plants."

11.5 N. M. Newmark and E. Rosenblueth, "Fundamentals of Earthquake Engineering," Prentice-Hall Inc., 1971.

11.6 ACI 349-01, "Code Requirements for Nuclear Safety Related Concrete Structures," Presented in ACI Manual of Concrete Practice, Part 4, 2001.

11.7 ACI 363R-84, "State-of-the-Art Report on High Strength Concrete," Presented in ACI Manual of Concrete Practice, Part 1, 1991.

11.8 ACI 435.2R, "Deflections of Reinforced Concrete Flexural Members," Presented in ACI Manual of Concrete Practice, Part 4, 1991.

11.9 Civil Design Guide 3DG-C01-00002, “Thermal Effects combined with Real Loads”

11.10 Civil Design Guide 3DG-C01-00017, “Loads and Load Combinations”




Figure 1: Finite Element Model for Framed-Type Structures(Including Mat Foundation)




Figure 2: Finite Element Model For Shear Wall Structures(Use Thin Shell/Plate Elements)




Figure 3: Finite Element Model For Shear Wall Structures(Use Solid Finite Elements)




Figure 4: Lumped Mass Stick Beam Model (Including Soil Spring and Dampers)




Rigid Links(Typ.)

Operating Deck

MAT (105' X105' X 7')

(51ft X 88 ft X9 ft)

8 ft X 8 ft(typ.)

El. 92.00 ft

El. 0.0 ftTop of Mat

Figure 5: Typical Coker Table Top Structure




165 f t

93 f t

Machine Unit #1

Machine Unit #2

Figure 6: Concrete Pedestal Supporting Two Units of Compressors

Documents

3DG-C01-00008 (1)