Introduction Connecting Mechanics to Analysis Connecting Analysis to Structural Design Theory of Structures Simplifying Assumptions Symmetries

Introduction to Structural Analysis

Mark A. Austin

University of Maryland

austin@umd.edu

ENCE 353, Fall Semester 2020

September 3, 2020

Introduction Connecting Mechanics to Analysis Connecting Analysis to Structural Design Theory of Structures Simplifying Assumptions Symmetries

Overview

1 IntroductionCourse Introduction

2 Connecting Mechanics to Analysis

3 Connecting Analysis to Structural DesignConnecting Analysis to Structural Design

4 Theory of StructuresStatically Determinate and Indeterminate Structures

5 Simplifying AssumptionsSmall Displacements, Linear Systems Behavior

6 Symmetries

Introduction Connecting Mechanics to Analysis Connecting Analysis to Structural Design Theory of Structures Simplifying Assumptions Symmetries

Introduction

Introduction Connecting Mechanics to Analysis Connecting Analysis to Structural Design Theory of Structures Simplifying Assumptions Symmetries

Definition of Structural Mechanics

Mechanics. Branch of science that deals with response of matterto forces.

Civil Engineering:

Structural mechanics (� - ✏):material displacement.

Geomechanics (� - ✏): pressure,temperature, displacements.

Fluid mechanics (� - ✏): pressure,velocities.

Other domains:

Biomechanics (� - ✏): eye, heart,biological systems that grow!

�

✏

Introduction Connecting Mechanics to Analysis Connecting Analysis to Structural Design Theory of Structures Simplifying Assumptions Symmetries

ConnectingMechanics to Analysis

Introduction Connecting Mechanics to Analysis Connecting Analysis to Structural Design Theory of Structures Simplifying Assumptions Symmetries

Pathway from Mechanics to System-Level Behavior

From material-level mechanics to building-system response:

�(x) =hM(x)EI

i

Material

Response

Section

Response

Beam

Response

Building System

Response

integrateintegrate assemble

✏(x , y)

Strain

�(x , y)

Stress Curvature

dy/dx

Slope

y(x)

Deflection

How will the integration work?

Analytical Procedures: The math needs to be ”nice” ...

Numerical Proedures: Compute approximate solutions !linear algebra, numerical algorithms, structural analysis andfinite elements.

Introduction Connecting Mechanics to Analysis Connecting Analysis to Structural Design Theory of Structures Simplifying Assumptions Symmetries

ConnectingAnalysis to Design

Introduction Connecting Mechanics to Analysis Connecting Analysis to Structural Design Theory of Structures Simplifying Assumptions Symmetries

Framework for Analysis and Design

Creating an Analysis Model

Abstract from considerationdetails not needed fordecision making.

Validate that modelcaptures essential aspects ofreal-world behavior.

Decision making needed fordesign.

Perfect is the enemy ofgood. Mathematical modeland decision making doesnot need to be perfect inorder to be useful.

Introduction Connecting Mechanics to Analysis Connecting Analysis to Structural Design Theory of Structures Simplifying Assumptions Symmetries

Connecting Analysis to Design

Structural Design. Sequence of analyses punctuated by decisionmaking.

Determine types and magnitudes of loads and forces acting onthe structure.

Determine context of project: geometric constraints,architectural constraints, geological conditions, urbanregulations, cost, schedule, etc.

Introduction Connecting Mechanics to Analysis Connecting Analysis to Structural Design Theory of Structures Simplifying Assumptions Symmetries

Connecting Analysis to Design

Generate structural system alternatives.

Analyze one or more of the alternatives.

Select and perform detailed design.

Implement/build.

Analysis and decision makingprocedures complicated byuncertainties in loading, materialproperties, etc. State-of-the-artmethods compensate foruncertainties with safety factors.

New structural systems may also require an experimental testingphase to verify behavior and achievable system performance.

Introduction Connecting Mechanics to Analysis Connecting Analysis to Structural Design Theory of Structures Simplifying Assumptions Symmetries

Connecting Analysis to Design

Real-World and Idealized Abstractions

Introduction Connecting Mechanics to Analysis Connecting Analysis to Structural Design Theory of Structures Simplifying Assumptions Symmetries

Connecting Analysis to Design

Introduction Connecting Mechanics to Analysis Connecting Analysis to Structural Design Theory of Structures Simplifying Assumptions Symmetries

Connecting Analysis to Design

Introduction Connecting Mechanics to Analysis Connecting Analysis to Structural Design Theory of Structures Simplifying Assumptions Symmetries

Theory of Structures

Introduction Connecting Mechanics to Analysis Connecting Analysis to Structural Design Theory of Structures Simplifying Assumptions Symmetries

Statically Determinate Structures

Definition. Can use statics to determine reactions and distributionof element-level forces. Determinacy is not a↵ected by details ofloading.

Two-Dimensional Problems

XFx

= 0,X

Fy

= 0,X

Mz

= 0. (1)

Three-Dimensional Problems

XFx

= 0,X

Fy

= 0,X

Fz

= 0. (2)

XM

x

= 0,X

My

= 0,X

Mz

= 0. (3)

Introduction Connecting Mechanics to Analysis Connecting Analysis to Structural Design Theory of Structures Simplifying Assumptions Symmetries

Statically Determinate Structures

Example 1. Simply supported beam:

A B

VB

VA

HA

PP

Three equations of equilibrium:P

Fx

= 0,P

Fy

= 0,P

Fz

= 0.

Three unknowns: VA

, HA

and VB

! Can use statics to solve.

Introduction Connecting Mechanics to Analysis Connecting Analysis to Structural Design Theory of Structures Simplifying Assumptions Symmetries

Statically Determinate Structures

Example 2. Small truss structure:

A B

VB

VA

HA

P P

PP

P

Use statics to find support reactions VA

, HA

and VB

.

Compute member forces by considering equilibrium ofindividual joints.

Introduction Connecting Mechanics to Analysis Connecting Analysis to Structural Design Theory of Structures Simplifying Assumptions Symmetries

Statically Indeterminate Structures

Definition. Statics alone are not enough to find reactions. Needto find additional information (e.g., material behavior).

Example 1. Simply supported beam:

VA

HA

VB

A B

HB

P P

Three equations of equilibrium:P

Fx

= 0,P

Fy

= 0,P

Fz

= 0.

Four unknowns: VA

, HA

, VB

and HB

! 4 > 3 ! staticallyindeterminate to degree 1.

Introduction Connecting Mechanics to Analysis Connecting Analysis to Structural Design Theory of Structures Simplifying Assumptions Symmetries

Statically Indeterminate Structures

Example 2.

VB

A B

VA

HA

VC

PP

Three equations of equilibrium. Four unknowns: VA

, HA

, VB

andVC

! 4 > 3 ! statically indeterminate to degree 1.

Example 3. Multi-material Truss Element.

Material behavior definedby � � ✏ characteristics.Need to maintaingeometric compatibility.

Material A Material B

P P