Chapter 1
Introduction-Concept of Stress
1.1
Review of Method of Statics
• Reading assignment
1.2
Stresses in a member
• In Statics (CE2450) we studied static equilibrium of rigid bodies
• In Mechanics of Materials (Strengths, CE3400) we will study static equilibrium if deformable
bodies, i.e. bodies that undergo a change in shape and/or size under the application of a force.
How can deforming body be in static equilibrium?
• Every structure is made up of structural components
.
x
FAB
F
B
B
F AB
P
P
X
B
X
A
A
dx
x
F AB
A
FAB
X
P
(FBD)
(b)
(a)
Figure 1.1: (a) Truss structure and (b) Force system acting on a component and parts of a
component
• Most structural components consist of
•• An axial direction x that is geometrically larger then the other dimensions, i.e. many structures are made of 1d components
1
2
CHAPTER 1. INTRODUCTION-CONCEPT OF STRESS
•• The other 2 dirs. will be quantified by areas (A) and moments of inertia (I, J), which are
also covered in the statics class
• Force F is external to the structure
• Force P is internal to the structural component. By defn. an internal force (to a component)
is the force exerted by one part of the component on the other to keep it in (static) equilibrium.
It could therefore be force exerted by BX on AX or the force exerted by AX on BX
•• Is FAB internal or external (or both)?
•• Even though for a truss component (axially loaded member or 2F member) the internal
force is a constant (w.r.t axial coordinate x), in the more general case we will have an internal
force system (forces and moments) that vary with x. In such cases the internal force system is
assumed to be constant for an infinitesimal element dx. Can you think of forces acting at A and
B that produce a non-constant internal reaction? (Hint: What happens if there is a shear force
VAB and moment MA /MB acting at A and B?)
• Principal focus of class is to determine deformation in 1d components (not necessarily trusses)
acted upon by a system of forces (at A and B)
• Intuitively we can see that deformation in the structural component AB depends on
•• Internal axial (normal) force as
Deformation∝ P and Deformation∝ A1
•• Axial⇒ along axial direction x, Normal⇒ normal (or perpendicular) to cut area (at X)
•• What kind of deformation does an axial stress produce?
• We will use “intensity” of internal force to measure deformation. The intensity of the axial/normal force is referred to as the normal/axial stress σ with
1
σ = PA . Note that we used PA1 is based on experimental observation of how typical materials
deform (change in size) due to an axial force
•• Units of σ are (same as
?)
N
3
P a(= m2 or Pascal), KP a(= 10 P a), M P a(= 106 P a), GP a(= 109 P a) in SI system
psi(= inlp2 or pounds per square inch), ksi(= 103 psi) in U S system
Intensity of axial internal forces represented as an average axial stress
.
1.2.
STRESSES IN A MEMBER
3
Ex.: Two solid cylindrical rods AB and BC are welded together at B as shown. Knowing that
d1 = 30 mm and d2 = 50 mm, find the average normal stress at the midsection of (a) rod AB
and (b) rod BC
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CHAPTER 1. INTRODUCTION-CONCEPT OF STRESS
Ex.: For the P ratt bridge and loading shown, determine the average normal stress in member
BE, knowing that the cross-sectional area of each member is 6 in.2 . How would you determine
the stress in member DE?
1.2.
STRESSES IN A MEMBER
5
Ex.: The frame shown consists of four 2 × 4-in. rectangular wooden members ABC, DEF , BE
and CF . Determine the maximum value of the average normal stress in member (a) BE and
(b) CF
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CHAPTER 1. INTRODUCTION-CONCEPT OF STRESS
Distribution of axial forces in a tensile member at mimimum area