Biaxial Analysis of General Shaped Base Plates
Biaxial Analysis of General Shaped Base Plates
Biaxial Analysis of General Shaped Base Plates
R. GONZALO ORELLANA 1
Summary: A linear model is used for the contact stresses calculation between a steel base
plate and a concrete foundation. It is also considered that the anchor rods tensions vary linearly.
Furthermore, strain compatibility is proposed. The non-linear system has as variables: the
neutral axis angle, the maximum anchor rods tension, the bearing length and optionally the
maximum stress in the concrete. The Newton-Raphson’s method is used to solve the system.
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similar to and concentric with the loaded
area.
(2)
(3)
(4)
(1) (5)
where, where,
= maximum design bearing strength, T = maximum tension force on anchor rods.
Section 10.14, ACI 318-08. a = maximum tension anchor rod area.
= compression resistance factor 0.65 by = maximum compressive stress.
Section 9.3, ACI 318-08. Y = bearing length.
= specified compressive strength of f = maximum anchor rod distance from NA.
concrete. n = elasticity ratio.
A1 = base plate area. (6)
A2 = maximum area of the portion of the = modulus of elasticity of steel.
supporting surface that is geometrically = modulus of elasticity of concrete.
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= compressive stress for node i+1.
To calculate the tension, in the anchor rod i, we = local coordinates for node i.
use a linear interpolation employing its local = local coordinates for node i+1.
coordinate y, as shown in Equation (7).
NEWTON-RAPHSON METHOD
This article explains briefly a method for a non-
(7)
linear equations system based on Newton-Raphson,
where, applied to a biaxial base plate problem case.
= tension in anchor rod i. The problem is to obtain variable values, such that
T = maximum tension force on anchor rods. the functions proposed are balanced.
f = maximum anchor rod distance from NA.
= local coordinate y of anchor rod i. (12)
The compressive concrete stress for node i of a We denote by X the entire variables vector, then
border segment is calculated by linear interpolation each function could be expanded by Taylor series.
as shown in Equation (8).
(8) (13)
where,
= compressive stress at node i.
= maximum compressive stress. Terms and higher are neglected. Then, we
obtain a set of four lineal equations to determine the
Y = bearing length.
= local coordinate y of node i. correction vector , which moves the functions
closer to zero simultaneously on each iteration. Since
The equivalent force, moment about local x axis the system converges, the term in the Equation (13)
and moment about local y axis for a line segment , then we get,
between node i and node i+1 can be determined by
the Equations (9), (10) and (11) respectively.
(14)
(10) (15)
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The next step is to determine the equivalent forces
(17) and moments in the local coordinate system for the
concrete compression and anchor rods tension.
donde,
= differential of variable j.
PRACTICAL APPLICATION
To continue with the numerical example, we
applied an external axial compression load of 90 kip,
a moment about global X axis of 300 kip·in and
another moment about global Y axis of 440 kip·in to
the base plate. The steel of the base plate has a
modulus of elasticity equal to 29000 ksi and the
modulus of elasticity of the supporting concrete is
3122.02 ksi.
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REFERENCES
1. Blodgett, Omer W., Design of Welded Structures,
1982.
2. Dake, Richard M. and Elkin, Sharon J., Beam-
Column Base Plate Design LRFD Method, 1999.
3. Rodríguez, J. A. and Aristizabal, J. Dario, Contact
Pressure Caused by a Rigid Footing of Any Shape
Subjected to Eccentric Axial Load, 2004
In Figure 4, the base plate to concrete contact
4. American Institute of Steel Construction (AISC)
stresses are shown. The steel anchor rods tension
Manual, 13th Edition, 2005
forces are displayed in Figure 5. 5. Base Plate and Anchor Rod Design, Steel Design
Guide 1, 2nd Edition, 2006
CONCLUSIONS
Since the developed procedure is an extension
of uniaxial methods, widely accepted for the
analysis of base plates, it also gives accurate
results for moments about one axis.
Another advantage, of the presented biaxial
procedure calculation, is to be an alternative to
the finite element method (FEM) for the analysis
of special projects with arbitrary base plate
shapes with openings. This method could be easily
abstracted into a computational algorithm.
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