The Direct Stiffness Method for Linear Static Analysis

In this lecture, we are going to talk about material stiffness and we will also get introduced to a stiffness formula that will be the heart of the direct stiffness method or the direct stiffness method or Finite Element Method.

this is the formula that gives the direct stiffness method the advantage that allows for more accurate structural analysis.

The direct stiffness method has several advantages: The formula for the element stiffness matrix is always the same for a given element type. Once the structural displacements are determined, the element stiffness matrix can be used to find the forces in each element.

This lecture will cover and set the stage for understanding upcoming terms and many other things that you know of but lacked the base idea to fully understand such as:

Understanding Young's Modulus

Difference between Stiffness and Flexibility

Understanding the Finite Element Method

Stiffness

Stiffness ✔

the definition of stiffness: Stiffness is defined as the resistance to a force causing a member to bend. Stiffness is very important to the end-use performance of many papers. For example, file folders and index cards must support themselves upright during use.

Knowing the sign convention for Joint Displacement and forces becomes crucial for the complex calculations of the direct stiffness method.

We will see terms similar to the following in this lecture:

Finite Element Analysis - Nodal Displacements, the Forces in each Element, and the Reactions

Finite Element Analysis - Nodal Displacements, Element Forces & Stresses, & the Support Reactions

Understanding the Finite Element Method

Finite Element Analysis - Nodal Displacements, Local Element Forces and Reaction Forces

check the FEM Digital Notes for quick revision

In this lecture will see what the stiffness matrix looks like for 2D Truss members in the local axis system.

In this section we will learn and cover:

Direct Stiffness Method for Truss Element

Analysis of Trusses Using Finite Element Methods | FEA Truss joints Methods | Structural Engineering

Finite Element Analysis on TRUSS Elements | FEM problem on trusses| Truss Problems in FEM

Finite Element Analysis - For the Truss shown, Solve for the Horizontal and Vertical Displacements

Finite Element Analysis - Determine the Displacement of Node 2 using the Stiffness Method

FEM TRUSS PROBLEM | Calculate the Nodal displacements| stiffness matrices| Finite Element Method

Finite Element Analysis - Solve for the Horizontal and Vertical Components of Displacement at Node 1

Analysis of TRUSSES With FE Analysis | Displacements Calculation for the truss elements

Finite Element Analysis - Determine the Displacement Components at Node 3 and the Element Forces

Finite Element Analysis - Determine the Horizontal and Vertical Displacement of Node

Analysis of TRUSSES With FE Analysis | Displacements  Calculation for the truss elements

Finite Element Analysis Example Problems | 2D trusses

SA48: Matrix Displacement Method: Truss Analysis

Finite Element Analysis Explained | Thing Must know about FEA Trusses

Truss Direct Stiffness Method

Direct Stiffness - 1D Truss

Problems on Analysis of Pin Jointed Trusses by Stiffness Matrix Method

Finite Element Trusses

Finite Element Method for Trusses

The direct stiffness method (DSM) is a method to solve statically determinant or indeterminant structures that is particularly well-suited for computer implementation.

Truss FEM

Featured snippet from the web

The simplest and most widely used finite element for truss structures is the well-known truss or bar finite element with two nodal points. Such kind of finite elements is applicable for analysis of the skeletal type of truss structural systems both in two-dimensional and three-dimensional space.

In this lecture, we are going to see how we can transform the local stiffness matrix from the previous lecture to the global stiffness matrix. it is essential to transform the local stiffness matrix into the global stiffness matrix when analyzing a truss with more than 1 member as well as frames and/or other structures.

This lecture can be related to the following:

Global and Local Stiffness Matrix Composition in 2D - Finite Element Methods

CE 383 - Direct Stiffness Method - Transformation between Local and Global Axes, Week 13

Easiest way of calculating Global Stiffness Matrix | global stiffness matrix assembly | 2021

How to Calculate the Global Stiffness Matrices | Global Stiffness Matrix method | Part-02

3 4 Global Stiffness Matrix for a Bar Element Arbitrarily Oriented in the Plane HD

Trusses (2/6): Transformation Matrix and the Global Element Stiffness Matrix

Intro to FEM - Week05-25 Frame Transformation Matrix

Development of Stiffness Matrix Problem 1 - Introduction to Stiffness Method in Matrix Form

TRUSS element: Transformation matrix and relation between local and global displacements / forces

Transformation of the stiffness matrix in global/reference coordinate system Payal Desai,Civil, NUV

Assembly of Global Stiffness Matrix(FEA)

Truss Direct Stiffness Method

Finite Element Method - Basics of obtaining global stiffness matrix

Trusses (4/6): Example - Transform Element Matrices and Assemble Global Stiffness Matrix

4. Stiffness of an inclined bar element | Derivation | Coordinate transformation

Intro to FEM - Week02-08 Truss Global Stiffness Matrix 01

1 TRansformation form global to local coordinate system sstem

TRUSS element transformation matrix

Intro to FEM - Week02-09 Truss Global Stiffness Matrix 02

Global stiffness matrix // Lecture 7 // Finite Element Method

Derivation of a Global Stiffness Matrix

Local Stiffness Matrix - an overview | ScienceDirect Topics

This lecture is the continuation of the last lecture.

Here we are going to see how we can transform the global stiffness matrix into the local stiffness matrix. This will further ensure the correctness of the derivation that we did in the previous 2 lectures.

This lecture is the most important lecture of this section, in this lecture, we will learn how to do FEA (Finite Element Analysis) of a 2D truss. we have already learned about the local and global stiffness matrix now we will apply those theories to assemble the stiffness matrix and load vector of a 2D truss and then use those to get the deflections and support reactions of the Truss.

The content of this lecture will cover the following:

Truss Direct Stiffness Method

Problems on Analysis of Pin Jointed Trusses by Stiffness Matrix Method

Analysis of Trusses Using Finite Element Methods | FEA Truss joints Methods | Structural Engineering

Calculate Nodal Displacements using Local and Global Stiffness Matrix EXAMPLE (Part 1 of 2)

Finite Element Analysis Example Problems | 2D trusses

Matrix stiffness method of Truss analysis

Lecture 10 - The Matrix Stiffness Method (2D Trusses)

SA49: Matrix Displacement Method: Frame Analysis (Joint Loads)

Analysis of TRUSSES With FE Analysis | Displacements & Stresses Calculation for the truss elements

Derivation for stiffness matrix for truss

Direct Stiffness Method for Truss

This is the continuation of the problem seen in the last lecture. in this lecture, we use the deflections found by FEA to calculate the reaction of the supports and how to use the FEA (Finite element analysis) to calculate the truss member forces.

Direct Stiffness - 2D truss example 1 conclusion

If we are doing FEA by hand then at times it is easier to get the stiffness matrix for the whole truss using equations instead of matrices, and that is what we are going to be observing in this lecture.

this is just to demonstrate the possibility of doing the FEA without matrix operation but is not recommended as for further use of FEA (Finite Element Analysis) you would need to learn FEM (Finite Element Modeling) for which you need computer programing or commercial software which all use the matrix operations to do FEA.

To move further in this course we need the ability to identify the zero-force members of a truss. Many of us might need a brief reminder of how to do that, and this lecture is for that purpose.

The contents of this lecture will cover the following:

How to identify zero force members in trusses

Statics: Lesson 50 - Trusses, How to Find a Zero Force Member

SA05: Truss Analysis: Zero-force Members

Understanding Zero Force Members Truss (Engineering Mechanics)

How to find Zero Force Members(with solved examples), OUR way

Find the zero force members in a truss: example #1

(5-39) Identifying Zero Force Members

Statics - Trusses - Zero force members

ST14: Identifying Zero-Force Members in Trusses

How to identify zero force members in a truss by inspection

FE Exam Statics - Zero Force Members On A Truss

Zero-Force members explained with an example

ZERO-FORCE MEMBERS IN TRUSSES | STRUCTURAL THEORY

Rules of Zero-Force Members

Trusses and zero force members

Engineering Statics | Theory | Truss Zero Force Members

Structures-Zero Force Members in Truss

how to identify zero force members

now we combine everything we have learned up till now to solve a problem of 2D Truss by hand using the FEA within 5 minutes without any computer program or algorithm or any assumption that hampers the quality of analysis.

Finite element analysis of 2D truss

identification of zero-force members

application of the direct stiffness method

analysis of 2D Truss

now we combine everything we have learned up till now to solve a problem of 2D Truss by hand using the FEA within 5 minutes without any computer program or algorithm or any assumption that hampers the quality of analysis.

Finite element analysis of 2D truss

identification of zero-force members

application of the direct stiffness method

analysis of 2D Truss

if we put together all of our Structural analysis knowledge and the FEA or the direct stiffness method we can solve for some 2D trusses by hand using FEA. No software packages or computer programs are required.

After learning the direct stiffness method for 2D trusses, now we will move on to 2D frames. and in frames there is shear and bending along with tension and compression which we have seen for trusses.  To handle these new forces we will modify the formula that we have been using but as you will see the formula will remain the same even after the modification. With that, we can also confirm that the direct stiffness method truly depends on stiffness, displacement, and forces and nothing else as the main base formula.

This Section will cover:

[International Lecture] STIFFNESS MATRIX METHOD FOR FRAME STRUCTURE AND ITS APPLICATION

Direct Stiffness - Frames 2d axial

Direct Stiffness Matrix Method for Analysis of Frames - Problem No 1

Direct Stiffness Method- Rigid Jointed Frames - Problem 1 - Structural Analysis 2

Direct Stiffness - Frames

Chapter 16-Frame Stiffness Matrix

Stiffness 04 - Analysis of Frames using Direct Stiffness Method

Direct Stiffness Method - Frame -1

Analysis of Frame using Flexibility Matrix Method - Problem No 1

Analysis of Frames by Stiffness Matrix Method - Problem No 2 ( Analysis of T Frame )

Lect:39-Stiffness Matrix Method for Frame Analysis

Lecture 28 : Matrix Method of Analysis: Frame (2D) (Contd.)

Stiffness matrix for a frame element in finite element analysis

Lecture 26 : Matrix Method of Analysis: Frame (2D)

Lecture 27 : Matrix Method of Analysis: Frame (2D) (Contd.)

Chapter 3: Stiffness Matrix Method [Frame]

SA50: Matrix Displacement Method: Frame Analysis (Member Loads)

Intro to FEM - Week02-09 Truss Global Stiffness Matrix 02

Stiffness Matrix Method for Analysis of Beams ( With Overhanging )

Stiffness Method for Frames - Part 1, Setup

Matching key words for this section:

Stiffness Method for Frames - Part 1, Setup

Matlab : Direct Stiffness Analysis of Statically Indeterminate Truss Part 1/2

Python: Direct Stiffness Analysis of Statically Indeterminate 2D Truss

Understanding the Finite Element Method

Lect:39-Stiffness Matrix Method for Frame Analysis

Direct Stiffness Method- Frame Analysis

CH5 Stiffness Matrix (Frame) Part 2/2

CE 483 - Frame Analysis with Direct Stiffness Method, Week 2

Stiffness Method | Working Rules | Direct Stiffness Method

Beam Elements Stiffness Matrices

Direct Stiffness Method - Frames

Direct Stiffness - 1D Truss

Frame Direct Stiffness 2

Analysis of Frames by Stiffness Matrix Method - Problem No 1

Matrix Method-Stiffness Method Of Structure Analysis

Stiffness Method for Frames - Part 1, Setup

Direct stiffness matrix with Excel VBA Part 3

Frame Analysis Example Using Matrix Stiffness Method

Direct Stiffness Matrix Method for Analysis of Beams - Problem No 1

Frame problem, direct stiffness method

Stiffness Method for Frames

Chapter 4: Direct Stiffness Method Truss Element [Skewed or Inclined Support Truss]

SAP2000 Stiffness Matrix

Problems on Analysis of Pin Jointed Trusses by Stiffness Matrix Method

Structural Analysis & Design III: Problem 2 (Frame) Stiffness Method (Direct) (Video 3)

Stiffness matrix for a frame element in finite element analysis

Stiffness Method | Working Rules | Direct Stiffness Method

Frames | Direct Stiffness Method | (Displacements, rotations & Internal forces ( BMD & SFD & NFD) )

direct stiffness plane frame

Multistorey Building Gravity Load Analysis-Stiffness Method

Chapter 3: Stiffness Matrix Method [Frame]

Frame Analysis Using Matrix Stiffness Method

Direct stiffness method, braced frame, SFD, example - Structural Engineering

Lecture 10 - The Matrix Stiffness Method (2D Trusses)

Analysis of beam element using direct stiffness method | Finite Element Method

Direct Stiffness Method: Portal Frame Part 1

SA48: Matrix Displacement Method: Truss Analysis

Analysis of frame by stiffness matrix method | Frame analysis | Sway Frame analysis | UET Peshawar

Direct Stiffness Method of Frames || Module5 || SA3 || KTU || S7 Civil

Direct Stiffness Matrix Method for Analysis of Frames - Problem No 2 ( Analysis of a Sway Frame )

Stiffness method Sway frame

Matrix stiffness method of Truss analysis

Direct Stiffness Method / Global Stiffness Matrix in Nepali Language with one solved example

KTU-Previous Year Questions - Direct stiffness Matrix (sway frame)

Direct Stiffness Matrix Method for Analysis of Frames - Problem No 3 ( Analysis of a Sway Frame )

Non Sway Frame Problem on Stiffness Method | Structural analysis - 2

#14 Stiffness Matrix Method For Frame || Part-2 || Structure Analysis-2 || In Nepali

Stiffness 05 - Analysis of Space Trusses & Space Frames using Direct Stiffness Method

Lecture 5 - Stiffness matrix of frame element - المهندس هشام الجمال

Direct stiffness matrix with Excel VBA Part2

FRAME ANALYSIS BY USING STIFFNESS METHOD

Frame Element Stiffness Matrices - Duke University

The Matrix Stiffness Method for 2D Frames - Duke People

the direct stiffness method for 2D frames

stiffness matrix for 2D Frame

TWO DIMENSIONAL ANALYSIS OF FRAME STRUCTURES ...

Matrix Stiffness Method Structural Analysis use Excel

Pin Jointed Truss Analysis By Stiffness Matrix Method। Lecture 10। Numerical। Structural Analysis 3

Direct Stiffness Approach - Introduction to Stiffness Method in Matrix Form

Matrix Stiffness Method for Analysis of Plane Frame

Stiffness matrix method for beam

Stiffness Method | Part 4 | Sway Frame | stiffness matrices method sway frame in Nepali

Portal Frame with Overhang. Analysis by Direct Stiffness Method. Part 2

Stiffness matrix method - Analysis of frame

Zahed Zisan - Civil Engineering & More Tutorials

Computation of Element Stiffness Matrix (2-D Frame)

SM 254 Matlab Tutorial #04. Stiffness method for Frame element

Structural Matrix Analysis - Member Stiffness Matrix

Concept Direct Stiffness Method - Structural Analysis 2

[1/3] Beam & Frame Analysis using the Direct Stiffness Method in Python | DegreeTutors.com

Lec-20 Direct Stiffness Matrix Method | Structure Analysis-2 | Part-1 | T.U.,K.U.,PO.U.,P.U,M.W.U

Structural Analysis & Design III: Problem 1 (Beam) Stiffness Method (Direct) (video 2)

Stiffness matrix method : sway analysis - G M Basha

Problem 1:Analysis of continuous beam using stiffness matrix method

Portal Frame with Overhang. Analysis by Direct Stiffness Method. Part 1

Direct Stiffness matrix // Lecture 6 // Finite Element Method

Stiffness Method Structural Analysis (Plane Frame Element)

Finite Element Analysis - Direct Stiffness Method I - Lecture

Analysis of frame by using stiffness method

Analysis of Frames by Stiffness Matrix Method - Problem No 4 ( Analysis of Sway Frame )

3D Space Frame Stiffness Method

In this lecture, we will see the transformation of the local stiffness matrix to the global stiffness matrix the same as we saw for the truss element in the truss section. Only this time for a frame element.

So,

In this lecture, we are going to see how we can transform the local stiffness matrix from the previous lecture to the global stiffness matrix. it is essential to transform the local stiffness matrix into the global stiffness matrix when analyzing a frame with more than 1 member as well as trusses and/or other structures.

In this example problem we will see the use of the stiffness matrix and the formula for direct stiffness analysis of 2D frames that we have seen in the last two video lectures.

this example problem will help us understand further the following things that are the learning outcome of this section:

• Stiffness matrix method for frame analysis

• Direct stiffness method for frames

• Matrix method of analysis for frames

• Frame stiffness matrix

• 2D frame analysis

• Structural analysis of frames

• Shear and bending in frames

• Modification of direct stiffness method for frames

the example problem itself is related to:

• Direct stiffness matrix method for analysis of frames - problem

• Analysis of frames using stiffness matrix method

• Direct stiffness method example

• Frame analysis example using matrix stiffness method

• Direct stiffness method for sway frames

• Analysis of sway and non-sway frames

with the end of this section we will have done the following:

• Direct stiffness method for LSA

• Linear static analysis

• Stiffness matrix method for structural analysis

• Matrix methods for structural analysis

• Truss and frame analysis

• 2D frame analysis with direct stiffness method

• Statically indeterminate truss and frame analysis

We will do the same example problem as the last lecture but this time we will neglect axial deformation. we will learn the following:

how and why can we neglect the axial deformation

if axial deformation can be neglected what does it mean for our analysis procedure and how can it help with our hand calculations?

Something not mentioned in the lecture is that :

the things that can help us with our hand calculation are not always good for developing a program for analysis using Python, C++, java, or any other programing language.  It is something that will be discussed in the future.

after understanding the FEA of frames and beams or the direct stiffness method for frames and beams and doing an example problem to understand the method further we will now do a practice problem to check and enhance our understanding.

the practice problem will be done using the matrix method in this lecture.

After doing Practice Problem 1, we now have an even better understanding of the method but because the support conditions were very favorable for the calculation some questions or some stones are left unturned in this topic and to cover that lacking or to get the answers to those questions we will do another practice problem which is similar to the first practice problem and yet different and more complex from the first problem.

with the example problem and the practice problems we have cleared all the basic understandings of the method and it can be quite similar to the 2D truss problems we have seen with the difference in the load calculations so to prevent any understanding issues or gaps regarding the joint load vector use we will go over the load part for FEA of frames and beams in this lecture again.

With the last lecture, we have a clear understanding of the FEA of frames and beams but that is using the matrix method which is suitable for both man and machine.

and now we will look into the equation method to do the same things that we did with the matrix method but this equation method is only suitable for humans and human language not machine or programming language.

We will do the same practice problem 1 of this section but this time we will do it using the equation method for better understanding and a comparison of the two ways of analyzing.

We will do the same practice problem 2 of this section but this time we will do it using the equation method for better understanding and a comparison of the two ways of analyzing.

with this lecture, we will start looking at how we can solve 3D truss problems using the direct stiffness method.

before we begin we need to know how to deduce the zero force members in a 3D truss system. Previously we have seen how to do that for 2D truss systems and this going to be something similar.

To do the analysis of 3D structures with the direct stiffness method we are in need computer programming as the hand calculation for these becomes quite difficult and time-consuming. But in this course, we are not looking into the programming part of FEA. That will come later but to get an understanding of the 3D structures analysis and also to look at some conditions where we don't need a computer program and can do the analysis by hand quite swiftly we will do a problem on 3D truss.

This section can be related to the following:

Finite Element Analysis - For the Space Truss, Determine the Nodal Displacements and Stresses

Finite Element Procedure Of 3D Truss System Using MATLAB

Analysis of TRUSSES With FE Analysis | Displacements & Stresses Calculation for the truss elements

Finite Element Analysis on TRUSS Elements | FEM problem on trusses| Truss Problems in FEM

3D Truss Reactions "By Hand"?

Finite Element Procedure Of 3D Truss System Using ANSYS

Lecture 33: Analysis of 3D Truss

Finite Element Analysis: L-06 Axial Truss Elements in 3D

Truss Element in Finite Element Methods | Truss Elements | Structural Analysis for Civil Engineering

FEM : Problem-based on trusses

Truss Direct Stiffness Method

Truss analysis by the method of joints: worked example #1

Chapter 14-Truss Stiffness Matrix (SI Units)

Direct Stiffness Method for Truss Element

Python: Direct Stiffness Analysis of 3D Truss

Stiffness 05 - Analysis of Space Trusses & Space Frames using Direct Stiffness Method

CH5 Stiffness Matrix (Truss) Part 1/2

Cornell Engineering - FEA Online Certificate

FEA Analysis Services - Experts of structural analysis

3D Truss Analysis - Duke People

Lecture 12: FE Modeling – 2D and 3D Trusses

Finite Element Analysis (FEA) of 2D and 3D Truss Structure

In this lecture, we will get acquainted with the stiffness matrix of the 3D truss nad we will learn how to memorize it and we will also see that it is quite similar to the stiffness matrix of the 2D truss. 

we will also learn how to calculate the angles between the members in 3D space.

after getting to know how to get the angle values, in this lecture, we will learn how to get the values of Cx, Cy, and Cz, without knowing the angular values.

this will come in handy when we do the example problem.

Now from the previous lecture we have learned how to calculate the angles between the members now in this lecture we will see how we can determine the length of the members using the value of the known angles.

from this lecture, we start the example problem of the 3D truss. in this lecture, we will determine the zero-force members of the 3D truss.

This lecture will cover and can relate to:

Zero Force Members in 3D Trusses

Zero Force Members for 3D Trusses Complete

How to identify zero force members in trusses

Statics: Lesson 50 - Trusses, How to Find a Zero Force Member, Method of Joints

SA05: Truss Analysis: Zero-force Members

How to find Zero Force Members(with solved examples), OUR way

Chapter 3-Space Truss

Understanding Zero Force Members Truss (Engineering Mechanics)

ST14: Identifying Zero-Force Members in Trusses

3D Truss Geometry Table Method

FE Exam Statics - Zero Force Members On A Truss

L3.3 Truss: concept of ZERO FORCE MEMBER || Examples || Engineering MECHANICS || FIRST YEAR course

Find Zero Force members with Zero Effort

3.3 Identifying Zero Force Members

simple trusses, the method of joints, & zero-force members

How to Identification of Zero Force Members in Truss

5.4 Zero-Force Members – Engineering Mechanics: Statics

Statics: Trusses

TRUSSES Learning Objectives 1). To identify zero-force members

In this lecture, we will learn how we can get the stiffness matrix for one member of a 3D truss for the practice problem. knowing how to get the stiffness matrix for one member will allow us to formulate a way to easily get the stiffness matrix for all the members and then formulate the global stiffness matrix for the entire truss system.

now that we know how to get the stiffness matrix for a single member we will use that knowledge to make a table that will allow us to get the stiffness matrix of the entire truss system within a couple of minutes without the use of any software or programming language. And once the table is ready we can easily solve the truss problem.