"Tables for the Analysis of Plates, Slabs and Diaphragms Based on the Elastic Theory" is a seminal engineering reference by Richard Bares
. It serves as a vital bridge between complex mathematical elasticity theory and the practical requirements of structural design. The Core Premise: Simplifying Complexity At the heart of the book is the Classical Thin Plate Theory
(often referred to as Kirchhoff-Love theory). Analyzing plates and slabs involves solving fourth-order partial differential equations (the Lagrange equation), which is notoriously difficult for everyday engineering practice. Bares’ work provides a comprehensive set of pre-calculated coefficients
that allow engineers to determine bending moments, shear forces, and deflections using simple arithmetic instead of advanced calculus. Key Components of the Analysis The tables are categorized based on three primary factors: Boundary Conditions:
Whether the edges are simply supported, clamped (fixed), or free.
Detailed analysis for rectangular and circular slabs, as well as more complex diaphragms. Loading Patterns:
Data for uniformly distributed loads, hydrostatic pressure, and concentrated point loads. Significance in Structural Engineering Before the ubiquity of Finite Element Method (FEM)
software, Bares’ tables were the industry standard. Even today, they remain essential for: Preliminary Design:
Quickly sizing structural elements before running complex computer simulations. Verification:
Providing a "sanity check" to ensure that software outputs are within a logical range. Educational Foundation: Helping students understand how different aspect ratios ( ) affect the distribution of internal forces in a slab. The Role of Elastic Theory By basing the tables on Elastic Theory
, Bares assumes that the material (usually reinforced concrete or steel) behaves linearly—meaning it returns to its original shape after loading and stress is proportional to strain. While modern design also considers "plastic" or "limit state" analysis, the elastic approach remains the primary method for ensuring serviceability
, such as preventing excessive cracking or deflection in floor systems. Conclusion
Richard Bares’ work transformed theoretical elasticity into a functional tool. By condensing thousands of hours of manual calculation into organized tables, he enabled a generation of engineers to design safer, more efficient buildings and bridges with high precision. or a specific coefficient table for a particular slab geometry?
The reference you are likely looking for is " Tables for the Analysis of Plates, Slabs and Diaphragms Based on the Elastic Theory
" by Richard Bareš. This seminal engineering handbook provides a comprehensive set of tables and formulas for calculating stresses and deformations in various flat structural elements. Overview of the Book
Purpose: Designed as a practical tool for design engineers to analyze structural components without requiring complex, from-scratch differential equation solving.
Scope: It covers rectangular and circular plates, slabs (plates loaded perpendicular to their plane), and diaphragms (plates loaded in-plane).
Theoretical Basis: The calculations are rooted in Classical Elastic Theory (typically Kirchhoff-Love plate theory for thin plates), assuming small deflections and linear elastic material behavior. Key Technical Contents
The handbook typically categorizes solutions based on the geometry and boundary conditions of the element: Rectangular Plates: Tables for various aspect ratios (
) and support conditions (e.g., all sides simply supported, clamped-clamped, or mixed conditions). Slabs: Focuses on bending moments ( ), twisting moments ( Mxycap M sub x y end-sub ), and shear forces.
Diaphragms: Focuses on in-plane stress distribution (plane stress theory). "Tables for the Analysis of Plates, Slabs and
Loading Conditions: Includes solutions for uniformly distributed loads, hydrostatic loads, and concentrated point loads. Digital Access and PDF Resources
Physical copies or digitized versions of this handbook can be found through the following platforms:
Internet Archive: Offers a digital version of the 1971 edition, titled Berechnungstafeln für Platten und Wandscheiben.
Scribd: Some users have uploaded full PDF versions of the 1979 English edition for viewing or download.
Google Books: Provides a preview and bibliographic details for the book. Basic Theory of Plates and Elastic Stability
Tables for the Analysis of Plates, Slabs, and Diaphragms based on the Elastic Theory
Introduction
The analysis of plates, slabs, and diaphragms is a crucial aspect of structural engineering, particularly in the design of buildings, bridges, and other infrastructure projects. The elastic theory provides a fundamental framework for understanding the behavior of these structural elements under various loads. This document presents a compilation of tables for the analysis of plates, slabs, and diaphragms based on the elastic theory.
Tables for Plate Analysis
The following tables provide solutions for various plate configurations and loading conditions:
Tables for Slab Analysis
The following tables provide solutions for various slab configurations and loading conditions:
Tables for Diaphragm Analysis
The following tables provide solutions for various diaphragm configurations and loading conditions:
References
This draft provides a basic outline of the types of tables that can be used for the analysis of plates, slabs, and diaphragms based on the elastic theory. The actual tables and solutions will depend on the specific problem and the desired level of accuracy.
Unlocking Structural Precision: A Guide to Richard Bareš’s "Tables for the Analysis of Plates, Slabs, and Diaphragms"
In the world of structural engineering, while modern Finite Element Analysis (FEA) software dominates the landscape, there remains a profound need for reliable, classical methods for verification and preliminary design. One of the most enduring resources in this field is
Tables for the Analysis of Plates, Slabs, and Diaphragms Based on the Elastic Theory Richard Bareš
Originally published in 1971, this 676-page compendium serves as a bridge between complex elastic theory and practical engineering application. The Core of the Elastic Theory Bareš’s work is rooted in the classical theory of thin plates Table 2: Deflections and Moments in Circular Plates
, which assumes small deflections relative to the plate's thickness. The analysis typically relies on the governing fourth-order partial differential equation:
nabla to the fourth power w equals the fraction with numerator q and denominator cap D end-fraction is the transverse deflection, is the distributed load, and is the flexural rigidity of the plate. Why This Resource Remains Essential
Even in an era of digital modeling, this handbook provides several critical advantages for engineers: Manual Verification
: Engineers use these tables to perform "sanity checks" on complex FEA results, ensuring that software outputs align with established elastic behavior. Rapid Preliminary Design
: For standard rectangular or circular slabs with common boundary conditions (pinned, fixed, or free), the tables allow for the immediate determination of moments and deflections without building a full digital model. Diverse Boundary Conditions
: The book covers a wide array of support scenarios for plates and diaphragms, including in-plane and out-of-plane loading. Comprehensive Scope : Beyond simple slabs, it includes analysis for diaphragms
(deep beams or wall-like structures) where in-plane stresses are dominant. Key Content Overview
The handbook is structured to guide a designer through various individual structural problems: Basic Theory of Plates and Elastic Stability
The analysis of plates, slabs, and diaphragms based on the elastic theory relies on mathematical models that describe how these structural elements deform under load. In structural engineering, designers often use standardized tables to bypass complex differential equations. These tables provide coefficients for moments, shears, and deflections based on boundary conditions and aspect ratios. Core Structural Elements
Plates: Flat structural entities where thickness is small compared to other dimensions. They primarily resist loads perpendicular to their surface through bending.
Slabs: A specific type of plate used in floors or roofs, often made of reinforced concrete. They can be one-way (supported on two sides) or two-way (supported on four sides).
Diaphragms: Structural elements that transmit in-plane lateral forces (like wind or seismic loads) to the vertical resisting elements, such as shear walls or frames. Theoretical Foundation: The Elastic Theory
The elastic theory assumes that the material returns to its original shape after the load is removed. For plates and slabs, this is typically governed by the Kirchhoff-Love plate theory, which makes several key assumptions:
The material is linear, elastic, homogeneous, and isotropic. The mid-plane remains unstrained during bending.
Plane sections remain plane and perpendicular to the neutral surface.
The thickness of the plate does not change during deformation. Understanding Analysis Tables
Engineers use tables—most notably those by Richard Bares or the Portland Cement Association (PCA)—to simplify the design process. These tables are organized by: Support Conditions: Simply supported edges. Fixed (clamped) edges. Free edges. Aspect Ratio ( ): The ratio of the long span to the short span. Load Distribution: Uniformly distributed loads (UDL). Hydrostatic (triangular) loads. Point loads. Key Formulas Derived from Tables
When using these tables, the internal forces are calculated using the following general forms: Bending Moment: Deflection: Shear Force: are coefficients from the tables, is the load, is the span, and is the flexural rigidity. Application in Diaphragm Analysis
Diaphragm analysis treats the floor or roof system as a deep horizontal beam. The tables help determine: Chord Forces: Tension and compression at the edges.
Shear Flow: Distribution of lateral forces across the diaphragm surface. Table 3: Deflections and Moments in Plates with Point Load
In-plane Deflection: Critical for checking the compatibility of deformations with vertical elements.
💡 Design Tip: While tables are excellent for regular shapes, complex geometries or irregular openings usually require Finite Element Analysis (FEA) software to ensure accuracy.
If you are looking for a specific set of values, I can help you find or calculate them if you provide: The dimensions of the slab (length and width)
The boundary conditions (e.g., all sides fixed, or two sides simply supported) The type of load (point load vs. uniform load)
The material properties (like the thickness and Poisson's ratio)
Given: ( a = 5m, b = 6m, h = 0.2m, E = 30 GPa, \nu = 0.2, p = 10 kPa )
First compute ( D = \frac30\times10^9 \cdot 0.2^312(1-0.04) = \frac30e9 \cdot 0.00812\cdot0.96 = \frac240e611.52 \approx 20.83 \times 10^6 , Nm )
Maximum deflection ( w_max = 0.00192 \cdot \frac10,000 \cdot 5^420.83e6 )
( 5^4 = 625 ), numerator ( 10,000 \cdot 625 = 6.25e6 )
( w_max = 0.00192 \cdot \frac6.25e620.83e6 = 0.00192 \cdot 0.30 \approx 0.000576 , m = 0.58 , mm )
Maximum moment ( M_max = 0.045 \cdot 10,000 \cdot 5^2 = 0.045 \cdot 250,000 = 11,250 , Nm/m )
This value is used directly for reinforcement design per meter width.
The tables are almost exclusively based on Kirchhoff–Love plate theory (classical thin plate theory), which assumes:
For slabs and diaphragms—where in-plane stiffness dominates for diaphragm action, and out-of-plane bending for slabs—the governing differential equation is the biharmonic equation:
[ \nabla^4 w = \fracpD ]
where ( w ) is the lateral deflection, ( p ) the load intensity, and ( D ) the flexural rigidity. Solving this equation analytically for arbitrary boundary conditions and loading patterns is mathematically intense, which is why precomputed tables are so powerful.
In the age of finite element software (FEA) like ANSYS, Abaqus, and SAP2000, one might assume that classical hand calculation methods have become obsolete. However, for the discerning structural engineer, the opposite is true. The elastic theory of plates remains the bedrock of structural intuition.
The search for the "Tables for the Analysis of Plates, Slabs, and Diaphragms Based on the Elastic Theory PDF" is not a search for outdated information; it is a search for reliability, speed, and fundamental understanding. These tables, popularized by the seminal work of theorists like Timoshenko, Czerny, and later Bares, provide closed-form solutions to partial differential equations that govern plate bending.
If you are designing reinforced concrete flat slabs, steel floor plates, or shear diaphragms, this resource is indispensable. This article explores where to find these tables, how to interpret them, and why they still outperform software in preliminary design.
By the 1990s, institutions like the U.S. Forest Products Laboratory, the Portland Cement Association (PCA), and European steel construction institutes began scanning their out-of-print table collections. Today, sites like Archive.org, Engineering Toolbox, and academic repositories host high-quality PDFs. However, due to copyright, many are still circulated privately or via university libraries.
