SoFA: A Matlab-Based Educational Software for the Shallow Foundation Analysis and Design KONSTANTINOS NIKOLAOU, DIMITRIS PITILAKIS Department of Civil Engineering, Aristotle University of Thessaloniki, Thessaloniki, Greece Received 21 June 2016; accepted 5 December 2016 ABSTRACT:
Shallow Foundation Analysis (SoFA) software is a newly-developed free stand-alone program based on Matlab for the calculation of bearing capacity and settlements of shallow foundations. SoFA uses several well-known formulas from the literature and design codes that are preferred in engineering practice and not complicated numerical methods. The main goal is to provide the end-user (student, academic, or professional civil engineer) with a simple, useful, user-friendly tool to perform preliminary assessment of foundation analysis. ß 2017 Wiley Periodicals, Inc Comput Appl Eng Educ; View this article online at wileyonlinelibrary.com/journal/cae; DOI 10.1002/cae.21791 Keywords: civil engineering; matlab; geotechnical; bearing capacity; settlements
INTRODUCTION Foundation is the part of the structure that transfers the loads applied to the superstructure safely to the ground. Footings are expensive and difficult to repair since they are constructed under the ground, so their design must be safe and at the same time minimizing the costs. Many different kinds of foundation types (i.e. pile, raft etc.) have been developed based on loading and soil conditions. Spread footings, and shallow foundations in general, are the ones most commonly used. Foundations are considered shallow if their depth to length ratio is small enough compared to their length. Shallow foundation design procedure consists of several checks (see for example [1–6]), among them the two most critical are: bearing capacity and settlement calculations. Modern computer software used in engineering practice are divided in two major categories: analytical, that use empirical formulas, and numerical that use finite element models to assess ultimate bearing capacity or settlements of footings [7]. Analytical solutions are preferred to numerical methods (i.e. complex finite element models) in engineering practice, mainly because of their inherent simplicity. SoFA [8] is a matlab [9] based software with a graphical user interface (GUI), that was developed within the framework of the post graduate courses (1) Engineering Seismology and (2) Soil Dynamics and Earthquake Resistant Design of Foundations,
Correspondence to K. Nikolaou (
[email protected]). © 2017 Wiley Periodicals, Inc.
Retaining Walls and Earth Structures. The aforementioned courses are two of the obligatory courses of the post-graduate study program Antiseismic Design of Structures of the Civil Engineering Department at the Aristotle University of Thessaloniki. SoFA allows for the calculation of bearing capacity and settlements of shallow foundations. It has a simple userfriendly graphical interface, is freely distributed and well documented in order to attract engineers to exploit its capabilities. A wide variety of educational software has been developed in the past for structural [10–18], geotechnical [19–24], and earthquake engineering [25–28] problems. In SoFA, safety factors against bearing capacity failure of the foundation soil for both strip and rectangular footings are calculated using several well-known formulas from literature. For static bearing capacity, Eurocode 7 [3], DIN 4017 [29], and the old Greek design code EAK2000 [30] are used. Moreover the literature formulas proposed by Meyerhof [31,32] and Hansen [33] are also included. For the bearing capacity under earthquake loads, the Eurocode 8 [34] and EAK2000 approaches are followed. Short-term components of settlements are calculated using the adjusted elasticity method [2], while confined one-dimensional deformation of the soil is assumed for the consolidation. SoFA provides solutions for all three design approaches implemented in Eurocode 7, as well as in Eurocode 8 for earthquake loading. Cohesive and cohesionless soils, static and dynamic loads, drained and un-drained conditions are examined. Extended reports of the calculations are given in textual and graphical form. It is not uncommon for different design codes to lead to completely different results. Bearing capacity is usually
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underestimated, while settlements are overrated, resulting to conservative foundation design. With SoFA all the results can be immediately compared using modern computer graphics capabilities. SoFA uses the concept of partial safety factors presented in Eurocode 7 in a fully customizable way. The user can modify every partial safety factor independently. Although the main purpose is to solve shallow foundation problems, bearing capacity formulas are of general interest and can be applied in raft foundation design or other bearing capacity problems that is retaining walls. It is important for engineering students to see beyond code provisions, since some design codes take into account crucial phenomena not described in others. Using SoFA differences between several design formulas are used to compute the static and earthquake load bearing capacity. Before using complicated finite element analysis models at least one preliminary solution should be obtained using simple analytical calculations, handbook formulas or trusted previous solutions. Some of this effort may lead to a better mathematical model to check computed results. This work has to be done before finite element analysis, rather than after.
THEORETICAL FRAMEWORK Several methods (i.e. theoretical or based on in situ tests) are used to calculate bearing capacity and settlements of a shallow footing. These methods use appropriate soil strength parameters, and geometry to assess the ultimate soil bearing capacity under static or dynamic loads, as well as footing settlements.
Static Bearing Capacity Calculation The general bearing capacity equation consists of the following three-term equation [i.e. 1,2]: . qu ¼ cN c sc ic bc gc dc þ p0 N q sq iq bq gq dq þ 1 2 BgN g sg ig bg gg d g ð1Þ The first term concerns cohesion (c) and is zero for cohesionless soil; qu is the bearing capacity of the foundation, p0 is the stress applied on the foundation depth, B is the foundation width, g is the soil weight, sj are the shape factors, ij are the load inclination factors, bj are the base inclination factors, gj are the soil inclination factors, and dj are factors related to the depth of
Figure 1 Different components contributing to bearing capacity of foundation soil.
foundation, with subscripts j¼c, q, g corresponding to cohesion, overburden pressure, and soil weight terms, respectively (see Fig. 1). All these factors vary in the different literature approaches and are given in detail in the appendix and in the software user’s manual [8]. Using different values for the factors, different footing shape, and drained or undrained loading conditions can be considered. Figure 2 shows the general geometry description of a problem considered in SoFA. Ly and Lx are width and length of the foundation, respectively, q is surcharge pressure, a is the foundation base inclination, b is the soil inclination angle, V is vertical (gravity) centered load, Hx is horizontal centered load, df is depth of foundation base, dw is depth of water table. SoFA calculates ultimate bearing capacity according to Eurocode 7 [3], the DIN4017 [29], the National Greek design code EAK2000 [30], and according to well-established and widely used solutions by Meyerhof [31,32] and Hansen [33].
Earthquake Bearing Capacity Calculation For earthquake loads, bearing capacity calculation is different from static loads. In SoFa two widely used engineering approaches are adopted. The one described in Eurocode 8 [34] and the one in the Greek design code (EAK 2000 [30]). According to Eurocode 8 (EN 1998, part 5—Annex F), bearing capacity is checked using the following formula: C C 1 eF T bV T F N; V; M; F ¼ b 0 a k k N 1 mF N
Figure 2 Geometrical input of the problem.
þ
C0 C 1 f F M gM M d 1 0 c k k N 1 mF N
ð2Þ
SOFA: A MATLAB-BASED EDUCATIONAL SOFTWARE
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Figure 4 Characteristic point for settlement calculation.
On the other hand EAK 2000 suggests using the same formulas with the static bearing capacity but with a different safety factor and a reduced angle of friction (for saturated soils), in order to approximately take into account the porepressure increase due to the earthquake.
Figure 3 Soil layers defined to calculate the settlements.
where: N ¼ g Rd
N Ed V Ed M Ed ; V ¼ g Rd ; M ¼ g Rd N max N max N max
ð3Þ
Settlements Calculation
The design axial force N Ed , shear force VEd, and bending moment M Ed are normalized using the ultimate vertical centered load Nmax the footing can bear. It is obvious that all normalized values are less or equal to 1. The normalized axial force must also fulfill the following 0 k k . F is the dimensionless inertia force constraint: N < 1 mF
Typically two types of settlements are calculated for the foundation design: immediate settlements, due to elastic deformation of the soil under static load, and those due to consolidation [2]. Immediate settlements of a footing are calculated by:
developed in the soil beneath the foundation, and g Rd a model partial factor varying between 1 and 1.5. In case of purely cohesive soils the ultimate vertical bearing capacity N max and factor F are:
where I is the shape and rigidity factor (values from tables, see [2,36,37]), p is the vertical footing pressure, v is the Poisson’s ratio, and E is the modulus of elasticity of the soil. To calculate the settlements due to consolidation, four sub-layers are defined beneath the foundation down to depth four times the foundation width. A characteristic point is selected in the middle of each layer, as shown in Figure 3, and soil stresses are computed for the initial state and due to the load case. Load stresses and Is factor are calculated using the formulas described in [2]. Integrating Boussinesq equation over a rectangle
N max ¼ ðp þ 2Þ
rag SB cu B; F ¼ cu gM
ð4Þ
where r is the unit mass of the soil, cu is the undrained shear strength of the soil and g M a partial material factor. S is the soil parameter defined in Eurocode 8 part 1 and ag the design ground acceleration. For cohesionless soils N max and F are given by:
ag S aV 2 g ð5Þ B Ng ; F ¼ N max ¼ 1 2 rg 1 g gtanw0 w where g is the acceleration of gravity, aV is the design vertical ground acceleration on type A soil (may be taken equal to aV ¼ 0:5ag SÞ. N g is the bearing capacity factor. Coefficients a–f, m, k, k’, cT, cM, c’M, b, g are provided in the Eurocode 8 [34] for purely cohesive and purely cohesionless soil conditions, whereas for intermediate conditions linear regression is used. The aforementioned Eurocode 8 formula was developed and is valid only for strip footings, but the formula can be applied to rectangular footings if the N max value is multiplied by (see [35]):
2e B ð6Þ sc ¼ 1 þ 0:32 1 B L
DH im ¼ IpB
1 v2 E
Figure 5 GUI input.
ð7Þ
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Figure 6 Graphical user interface of the SoFA software.
of dimensions B x L (Newmark, 1935 [37]) produces: pffiffiffiffi pffiffiffiffi 1 2MN V V þ 1 2MN V þ tan1 ð8Þ qV ¼ q0 I S ; I S ¼ 4p V þ V 1 V V V1 where M ¼ B=z; N ¼ L=z; V ¼ M 2 þ N 2 þ 1; V 1 ¼ ðMN Þ2 , I S becomes one for zero depth (z ¼ 0 m) and is calculated for point
A (Fig. 4), as the sum of I S factors calculated for the four rectangular areas defined by A. Settlements due to consolidation are then calculated at every characteristic point in Figure 3, using the following equations (see i.e. [2]):
Figure 7 Left: safety factor against bearing capacity failure comparison with five different approaches. The black dashed line denotes the indicative safety factor of 1.96. Right: breakdown of the ultimate bearing capacity with respect to contribution of overburden pressure, foundation geometry, and cohesion.
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The total settlement due to consolidation is the sum of the settlements for each layer. The total settlement of the foundation is the sum of the immediate and the one due to consolidation.
SOFTWARE DESCRIPTION In this section the software GUI and the website will be described. Some screenshots of the software input and output are presented.
Graphical User Interface
Figure 8 Graphical output of settlements (immediate, consolidation, and total) with depth.
0
0:1CC s OCR log 1 þ e0 s 0 init 0
CC s tot log 0 þH i 1 þ e0 s OCR 0
0:1CC s tot < s 0 OCR ; DH i ¼ H i log 0 1 þ e0 s init
Ifs 0 tot > s 0 OCR ; DH i ¼ H i
IFs 0 tot
ð9Þ ð10Þ
Where s0 tot is the effective shear stress at the mid-height of layer I, s’OCR is the effective over-consolidation stress, s0 init is the initial effective stress, Cc is the compression index, e0 is the gap percentage.
Figure 5 depicts, in the form of a simplified flow chart, the optional and required input for the GUI, in the order they should be introduced. Figure 6 shows the SoFA main GUI. This interface consists of several panels describing geometry of the problem, soil properties, loads, and data for settlement calculation. The user has to input all the required parameters and select the soil and loading type (i.e. cohesive or cohesionless/drained or undrained), design approach and if earthquake formulas are going to be considered. The Geometry of the problem panel is the input geometry of the footing, the depth of the foundation df, the depth of the water level dw, and the angle of the footing level a and angle of soil surface b with respect to the horizontal. The user can choose between two options for the footing geometry, namely rectangular or strip foundation. In the Soil properties panel the user chooses between soil type (cohesive or non-cohesive) and loading conditions (drained or undrained) from the drop-down menus, and accordingly the soil parameters are filled in. Partial factors can be applied to the cohesion and friction angle values (default partial factors ¼1). In the Loads panel, the user initially chooses the design approach according to Eurocode 7 [3], notably 1, 2, or 3. Following, the user provides the characteristic (unfactored)
Figure 9 Text report generated containing all calculations of bearing capacity and settlements.
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Figure 10
Software website (http://sofasoftware.weebly.com/).
values of loads (vertical, horizontal, and moment) at the geometrical center of the foundation and at the foundation-soil interface level, along with the partial load factors (by default equal to unity). In the Settlement Calculation Data panel, the user provides all parameters required for the settlement calculation, immediate and because of consolidation, whenever the latter one applies. In case bearing capacity has to be checked against earthquake loads, the user chooses yes in the earthquake design drop-down menu in the Loads panel, and provides necessary information for the calculations. Eurocode 8 Annex F, requires further information for the calculation of the earthquake bearing capacity (see Equations 2–6), safety factor gRd, ultimate vertical load Nmax and seismic inertia force F can be either entered by the user or automatically calculated by SoFA using the relevant button.
Results are produced when one of the relevant buttons is pressed (Bearing Capacity, Settlements, Report). An additional button to report bugs was introduced in order to make debugging easier. This button opens a web browser and directs it to the bug submission form of SoFAs’ website. Help button opens the users’ manual [8]. A clear button replaces all values with zeros. Graphical output for bearing capacity safety factors and ultimate capacities in kPa is shown in Figure 7 and for settlements immediate and due to consolidation in Figure 8. All figures can be saved as Matlab figure files, or can be exported as high resolution images to be inserted in word processing software. Text reports (Fig. 9) include all the intermediate calculations, and the relevant text files are immediately created on the working directory. A stand-alone executable file was created with the use of Matlab Compiler (MCR) in order to allow for users without a Matlab license to install and run the software on different
SOFA: A MATLAB-BASED EDUCATIONAL SOFTWARE
operating systems. This way SoFA software can be redistributed very easily and freely, along with the required Matlab component, which is included in the pre-compiled installation file.
Website Together with the release of the first version of SoFA, a modern and easy to use website was created (http://sofasoftware.weebly. com/ see Fig. 10). The website includes various useful pieces of information: user-registration forms, bug report forms, documentation, download, blog, mailing list tools etc. A trial version of SoFA (without the textual reports) is available for everyone, while for the full version registration is required. Up to now (November 2015), more than 1500 users worldwide have registered for a full version. It is interesting that a significant number of registered users are small design firm engineers. A detailed users’ manual, a quick introductory guide, and a solvedexamples report were also released through the website to document the software in the most efficient way possible.
EDUCATIONAL FRAMEWORK SoFA is designed to be used by engineers in practice, but mainly in academia as an educational tool. It is currently widely used in both undergraduate and graduate courses of the Civil Engineering Department of the Aristotle University of Thessaloniki. More than 200 undergraduate students per year use SoFA to validate hand calculations in their Foundation Analysis and design semester project, which produces valuable feedback in terms of different aspects, from installation of the software to theoretical solution issues. During the Foundation Analysis and Design course, SoFA is presented to the students, who are welcome to contribute in its developing with fruitful ideas. By using SoFA, students can understand in a user-friendly and interactive way the significant differences that exist among different foundation design methods, while they are able to use software results in their semester project. This way, SoFA is not used as a black box, but rather as a tool. Software current limitations (for example the uniform soil stratum) led to some creative comments from the students and an interest in participating in the further development of this software. So, SoFA it is not limited to teaching students foundation engineering but also encourages them to join software development teams. In order to receive some feedback, some forms were sent to the registered users, and also students during their oral course examination were invited to submit their comments. The feedback from users was very positive, and mainly concentrated on adding new features in future versions. Problems reported were mainly related to Matlab Runtime Compiler installation procedure (similar comments were reported in [10,11]).
CONCLUSIONS In the present work, SoFA, a recently-developed simple program based on Matlab and freely available online for analysis and design of shallow footings is presented. SoFA was developed at the Civil Engineering Department of Aristotle University of Thessaloniki and is also used as a teaching tool for undergraduate and postgraduate students. The main idea was to implement analytical formulas from the literature and design codes in order to
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allow for quick estimation of the bearing capacity and settlement of shallow footings. This way significant differences that exist among various design approaches are highlighted without the use of complicated numerical models. SoFA, as educational tool, is a very promising and has attracted students, educators and engineers worldwide, providing positive feedback and requesting additional features. A newer improved version including more design checks and other extensions will be released soon. Some online videos to make installation procedure easier will also be introduced since problems were reported in the setup of Matlab Runtime Compiler. Moreover, SoFA motivated the creation of a small software developers community at the Civil Engineering department of our University that will allow spreading the importance of new technologies applied in education and engineering practice.
ACKNOWLEDGMENTS The authors would like to acknowledge the help of Professor Christos Anagnostopoulos and Associate Professor Konstantinos Georgiadis during the development of the software. Their comments are greatly appreciated. The authors also thank Dr. Ivan Kraus from the University of Osijek for proofreading this paper and for his feedback.
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BIOGRAPHIES Konstantinos Nikolaou received his PhD in 2015 from the Department of Civil Engineering of Aristotle University of Thessaloniki, Greece. He obtained his MSc in Earthquake Engineering and a Civil Engineering degree from the same University. His areas of research include the direct methods of plasticity, nonlinear optimization techniques, geotechnical engineering, and software development.
Dimitris Pitilakis is assistant professor in the Department of Civil Engineering of Aristotle University of Thessaloniki, Greece (MSc University of California, Berkeley, PhD in earthquake engineering from Ecole Centrale Paris, France). He is an expert in geotechnical earthquake engineering, with emphasis on soil–foundation–structure interaction, dynamics of foundations, and performance-based geotechnical design. He is member of national and international scientific societies on Earthquake Engineering and reviewer of international scientific journals. He has developed scientific software for the simulation of the soil–foundation–structure interaction, with emphasis on nonlinear soil behavior, as well as software for foundation design and analysis. He has significant experience in experimental soil–foundation– structure interaction in small-scale (shaking table and centrifuge) and fullscale (EuroProteas in Euroseistest http://euroseisdb.civil.auth.gr/sfsis).
