MAE314 Solid Mechanics Fall, 2008

Dr. Yuan

1-1

Chapter 1. Introduction to Mechanics of Materials

1.1 What is Mechanics of Materials?

Objectives : To study mechanical behavior of solid bodies subjected to various types

of static loading.

Approaches : Develop analysis techniques (models) from basic principles

Structures : bar, beam, column, shaft, pressure vessel.

Issues : Strength, stiffness, stability.

Ultimate Goal : Design Safe Structures

Modeling is a process of deliberately neglecting the insignificant phenomena which

consciously capturing the important ones.

Mechanics

(A) Mechanics of Rigid Bodies

Static and dynamic behavior of undeformable bodies under external forces and/or

moments

(B) Mechanics of Deformable Solids

Internal forces, moments (stresses) and associated change in geometry of the

bodies (strains, displacements) under external forces and/or moments

Strength : determine whether the bodies will fail in service

Stiffness : determine whether the amount of deformation will be acceptable

Stability : determine whether the structure will collapse

These equations of statics are directly applicable to deformable bodies. The deformations

tolerated in engineering structures are usually negligible in comparison with the overall

dimensions of structures (except stability). Therefore, for the purposes of obtaining the forces in

members, the initial undeformed dimensions of members are used in the analysis.

1.2 The Fundamental Equations of Deformable-Body Mechanics

In general, three types of equations are used to solve deformable body problems

A. Equilibrium equations need to be satisfied;

B. Geometry of deformation must be prescribed; (boundary conditions, geometric

compatibility, etc.)

MAE314 Solid Mechanics Fall, 2008

Dr. Yuan

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C. Type of materials needs to be known.

1.3 Problem-solving Procedures

For a given problem, three steps are used to solve the problems correctly:

A. Plan the solution

: plan strategy and steps in solving the problems

B. Carry out the solution

: find the proper tools to solve the problems using equilibrium

equations, geometry of deformation, and material type.

C. Review the solution

: Does the dimension make sense? Are the quantities in a reasonable

manner including sign and magnitude? Does the solution violate the assumption you assumed

before solving the problem?

1.4 Review of Static Equilibrium; Equilibrium of Deformable Bodies

External and Internal Forces and Moments

Forces or moments that are applied to a structure are described as external (e.g., a weight

attached to the end of a rope). Forces and moments that are developed within a structure in

response to the external force systems present in the structure are described as internal (e.g., the

tension in a rope resulting from the pull of an attached weight).

(a) Free-body diagram of block

(b) Free-body diagram of beam

(c) Free-body diagram of beam

Fig. 1 Reactions and free-body (or equilibrium) diagrams. The weight of the beam has been

neglected in Fig. 1c.

The external forces systems

can be categorized into applied and reactive forces. Applied

forces are those that act directly on a structure (e.g., snow). Therefore, the applied forces move

MAE314 Solid Mechanics Fall, 2008

Dr. Yuan

1-3

as the structures deform. Reactive forces are those generated by the action of one body on

another and hence typically occur at connections and supports. Consequently, the reactive forces

support the structures and do not move. The existence of reactive forces follows from Newton's

third law, which generally states that whenever one body exerts a force on another, the second

always exerts on the first a force which is equal in magnitude, opposite in direction, and has the

same line of action shown in Fig. 1.a. In the figure 1.b, the force on the beam causes downward

forces on the foundation and upward reactive forces are consequently developed. A pair of action

and reaction forces thus exist at each interface between the beam and its foundation. In some

cases like Fig. 1.c, moments form part of the reactive system as well. The diagrams in the Fig. 1

which show the complete system of applied and reactive forces acting on a body, are called free-

body diagrams. The reactive forces and moments are presented as arrows with a slashed symbol.

In later chapters, the reactive forces and moments are drawn in single arrows for simplicity.

In broad sense, a free body diagram is a pictorial representation often used by physicists and

engineers to analyze the forces acting on a free body

. It shows all contact

and non-contact

forces

acting on the body.

After establishing the nature of the complete force system consisting of both applied and

reactive forces acting on the structure, the next step is to determine the nature of the internal

forces and moments developed in the structure as a consequence of the action of the external

forces.

Internal forces and moments

are developed within a structure due to the action of the

external force system acting on the structure and serve to hold together, or maintain the

equilibrium of, the constituent elements of the structure.

If the solid body as a whole is in equilibrium, any part of it

must be in equilibrium. For

such parts of a body, however, some of the forces necessary to maintain equilibrium must act at

the cut section. These considerations lead to the following fundamental conclusion: the

externally applied forces to one side of an arbitrary cut must be balanced by the internal forces

developed at the cut.

Fig. 2 The six internal forces and moments on an arbitrary cross section of a slender member.

MAE314 Solid Mechanics Fall, 2008

Dr. Yuan

1-4

Equilibrium Equations: (Using free body diagrams, FBD)

0,0,0

x y z

F F F

=

= =

∑ ∑ ∑

0,0,0

x y z

M M M

=

= =

∑ ∑ ∑

Support Reactions

1.5

A Short Review of the Methods of Statics

Two-bar structure: Points A, B, and C are hinged (no moment)

MAE314 Solid Mechanics Fall, 2008

Dr. Yuan

1-5

What is the two-force member

?

Force is along the direction of the member. Otherwise, it cannot satisfy the equilibrium.

Example 1.2.1

Free body diagram (FBD)

BY looking the original beam configuration, F

BC

is in tension, F

AB

is in compression.

Two equations are enough to determine all the

internal forces by using summing the forces

0=

∑

x

F

,

030

5

3

=−

BC

F

,

kNF

BC

50=

0=

∑

y

F

,

0

5

4

=−

ABBC

FF

,

kNF

AB

40=

The internal forces can be determined from the

moment equilibrium:

∑

= 0

A

M

,

0

5

4

60030800 =×−×

BC

F

,

kNF

BC

50=

∑

= 0

C

M

,

060030800 =

×

−

×

AB

F

,

kNF

AB

40=

The question: Can bars AB and BC sustain the load? Are they safe?

Problem 1.4-11 (a) determine the reactions at A and D, and (b) determine the internal resultants

(axial force, shear force, and bending moment) on the cross section at B.

MAE314 Solid Mechanics Fall, 2008

Dr. Yuan

1-6

Ans: A

x

= 0, A

y

= 5w

0

L/27, D

x

= 0, D

y

= 4w

0

L/27

(b) F

B

= 0, V

B

= 11w

0

L/108, M

B

= 17w

0

L

3

/324.

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