Mathcad deflection by meam | Manuais, Projetos, Pesquisas Engenharia de Software

Proceedings of the 2004 American Society for Engineering Education Annual Conference & Exposition Copyright

2004, American Society for Engineering Education

Session 2649

Use of Mathcad in Computing Beam Deflection by Conjugate Beam Method

Nirmal K. Das

Georgia Southern University

Abstract

The four-year, ABET-accredited Civil Engineering Technology curriculum at Georgia Southern

University includes a required, junior-level course in Structural Analysis. One of the topics

covered is the conjugate beam method for computing slope and deflection at various points in a

beam. The conjugate beam method is a geometric method and it relies only on the principles of

statics. The usefulness of this method lies in its simplicity. The students can utilize their already

acquired knowledge of shearing force and bending moment to determine a beam’s slope and

deflection.

An approach to teaching this important method of structural analysis that complements the

traditional lecturing through inclusion of a powerful, versatile and user-friendly computational

tool, is discussed in this paper. Students will learn how to utilize Mathcad to perform a variety

of calculations in a sequence and to verify the accuracy of their manual solutions. A Mathcad

program is developed for this purpose and examples to illustrate the computer program are also

included in this paper. The integration of Mathcad will enhance students’ problem-solving

skills, as it will allow them to focus on analysis while the software performs routine calculations.

Thus it will promote learning by discovery, instead of leaving the student in the role of a passive

observer.

Introduction

With the objective of enhanced student learning, adoption of various instructional technology

and inclusion of computer-aided problem-solving modules into the curriculum has been a trend

for civil engineering and civil engineering technology programs. More specifically, the effective

incorporation of a variety of software packages for the teaching-learning process related to the

structural analysis course has been addressed in several articles1,2,3,4,5,6,7,8 in recent years.

Analysis of both statically determinate and statically indeterminate structures, by classical

methods (slope-deflection and moment distribution) and stiffness method, using EXCEL,

MATLAB and Mathcad9, have been covered in those articles. However, one very important and

useful method, the conjugate beam method, was not addressed. The purpose of this paper is to

present a simple and effective approach used by the author to teach this important topic of

structural analysis incorporating the use of Mathcad software.

Page 9.1354.1

Mathcad deflection by meam, Manuais, Projetos, Pesquisas de Engenharia de Software

Documentos relacionados

Pré-visualização parcial do texto

Baixe Mathcad deflection by meam e outras Manuais, Projetos, Pesquisas em PDF para Engenharia de Software, somente na Docsity!

Session 2649

Use of Mathcad in Computing Beam Deflection by Conjugate Beam Method

Nirmal K. Das

Georgia Southern University

Abstract

The four-year, ABET-accredited Civil Engineering Technology curriculum at Georgia Southern

University includes a required, junior-level course in Structural Analysis. One of the topics

covered is the conjugate beam method for computing slope and deflection at various points in a

beam. The conjugate beam method is a geometric method and it relies only on the principles of

statics. The usefulness of this method lies in its simplicity. The students can utilize their already

acquired knowledge of shearing force and bending moment to determine a beam’s slope and

deflection.

An approach to teaching this important method of structural analysis that complements the

traditional lecturing through inclusion of a powerful, versatile and user-friendly computational

tool, is discussed in this paper. Students will learn how to utilize Mathcad to perform a variety

of calculations in a sequence and to verify the accuracy of their manual solutions. A Mathcad

program is developed for this purpose and examples to illustrate the computer program are also

included in this paper. The integration of Mathcad will enhance students’ problem-solving

skills, as it will allow them to focus on analysis while the software performs routine calculations.

Thus it will promote learning by discovery, instead of leaving the student in the role of a passive

observer.

Introduction

With the objective of enhanced student learning, adoption of various instructional technology

and inclusion of computer-aided problem-solving modules into the curriculum has been a trend

for civil engineering and civil engineering technology programs. More specifically, the effective

incorporation of a variety of software packages for the teaching-learning process related to the

structural analysis course has been addressed in several articles

in recent years.

Analysis of both statically determinate and statically indeterminate structures, by classical

methods (slope-deflection and moment distribution) and stiffness method, using EXCEL,

MATLAB and Mathcad

, have been covered in those articles. However, one very important and

useful method, the conjugate beam method, was not addressed. The purpose of this paper is to

present a simple and effective approach used by the author to teach this important topic of

structural analysis incorporating the use of Mathcad software.

Page 9.1354.

Conjugate Beam Method

Structures deform when subjected to loads, and a vast majority of structures undergo elastic

deformations only, under service loads. For linear elastic behavior, the Principle of

Superposition remains valid. Thus load effects (slope, deflection etc.) due to different types of

loads can be combined to obtain the final results.

The conjugate beam method is based on consideration of the geometry of the deflected shape of

a beam. A conjugate beam is a fictitious beam of the same length as the actual beam, but its

supports (as well as internal connections) are such that if the conjugate beam is loaded with the

M/EI diagram of the real beam, the shearing force and bending moment at any point on the

conjugate beam are equal, respectively, to the slope ( ) and deflection ( ) at that point of the real

beam. M is the bending moment and EI represents the flexural rigidity of the beam, where E is

the modulus of elasticity of beam material and I is the moment of inertia of beam cross-section.

The basis of the method is that the relations among load, shear and bending moment in a beam

are similar to the corresponding relations among M/EI, slope and deflection of the beam. The

application of the laws of equilibrium on a differential element of a beam leads to a pair of

equations relating the load, shear and moment. Likewise, integration of the governing

differential equation of elastic beam theory, expressing the moment-curvature relationship at a

point, leads to a pair of equations relating M/EI, slope and deflection. These derivations can be

found in any standard textbook on structural analysis10,11,12.

Advantages of Mathcad

Mathcad , an industry-standard calculation software, is used because it is as versatile and

powerful as programming languages, yet it is as easy to learn as a spreadsheet. Additionally, it is

linked to the Internet and other applications one uses everyday.

In Mathcad , an expression or an equation looks the same way as one would see it in a textbook,

and there is no difficult syntax to learn. Aside from looking the usual way, the expressions can

be evaluated or the equations can be used to solve just about any mathematics problem one can

think of. Text can be placed anywhere around the equations to document one’s work. Mathcad ’s

two- and three-dimensional plots can be used to represent equations graphically. In addition,

graphics taken from another Windows application can also be used for illustration purpose.

Mathcad incorporates Microsoft’s OLE 2 object linking and embedding standard to work with

other applications. Through a combination of equations, text, and graphics in a single worksheet,

keeping track of the most complex calculations becomes easy. An actual record of one’s work is

obtained by printing the worksheet exactly as it appears on the screen.

Program Features

The program developed by the author will require input data pertaining to the geometry of the

problem, material property and the loading. More specifically, the following information is

required as input data: beam type (simply-supported, simply-supported with overhang and

cantilever), length, moment of inertia and modulus of elasticity, magnitudes and lengths of Page 9.1354.

Example 1: Determine the slopes at ends A and D and the deflections at points B and C of

the beam shown in Figure 1. Use E = 1,800 ksi and I = 46,000 in