Chapter 1 the fundamental concepts

Shear stress: The intensity of force acting tangent ∆F x , ∆F y to ∆A is called the shear stress... determine the average normal stress developed in section a–a and in eachof the bolt sh

THE FUNDAMENTAL CONCEPTS

a Reliability

A force exerted on a system or on some of its components by an agencyoutside the system.

2.2.1 Concentrated force:

2.2 Classification of force

Loads Support Reaction

External loads

2.2.2 Distributed force:

 

q s Q

+ Linear distributed load

q(z) – magnitude of linear distributed force

Units: [force]/[length] N/m or kN/cm

2.2.2 Distributed load

+ Linear distributed force

2.2.2 Distributed force

q

L

γ(x, y, z) – magnitude

Unit: [force]/[length]3 N/m 3 or kN/cm 3

+ Body force

2.2.2 Distributed load

2.3.1 Smooth support

N

One unknown The reaction is a force which acts perpendicular to the surface at the point of contact.

Y X

X

Y

X M

L

0

0 0

 

 

 

000( ) 0( ) 0( ) 0

The equations of equilibrium

M

The equations of equilibrium

The equations of equilibrium

x

The equations of equilibrium

H

internal force acting in the section a-a.

12 mm

400 N

10 mm

30 mm

Problem Determine internal force acting in the section A-B

Problem The clamp is made from members AB and AC, which are pin

connected at A If it exerts a compressive force at C and B of 180 N, determine internal force acting in the section a-a

Problem The offset link supports the loading of P =180 N, determine

internal force acting in the section a-a

15mm

Problem The joint is subjected to a force of P = 80 lb and F = 0

Determine internal force acting in the section a-a

Problem The joint is subjected to a force of P = 80 lb and F = 120 lb

Determine internal force acting in the section a-a

Problem. If the baby has a mass of 5 kg and his center of mass is at G Determine internal force acting in the section a-a There are two rods,

one on each side of the cradle

Problem Determine internal force acting in the section a

Problem Determine internal force acting in the section A

Problem Determine internal force acting in the section a-a

C.

A, B and C.

C and D.

force, shear force, and moment at points F and H in the floor crane.

normal force, shear force, and moment at point A.

∆F – very small internal force acting on ∆A

∆Fx, ∆Fy, ∆Fz – components of ∆F in coordinates x, y, z

4.2 Shear stress: The intensity of force acting tangent ∆F x , ∆F y to ∆A is

called the shear stress

P A

 

A – cross-sectional area of the bar where σ ave is determined

P A

 

have cross-sectional areas of 10 mm2 and 15 mm2, respectively.

Determine the intensity q 0 of the distributed load so that the averagenormal stress in each rod does not exceed 300 kPa

0

q

the load has a mass of 50 kg The diameter of rod AB is 8 mm.

determine the average normal stress developed in section a–a and in each

of the bolt shanks at B and C Each bolt shank has a diameter of 12 mm.

pipe has an inner diameter of 20 mm and outer diameter of 28 mm Therod has a diameter of 12 mm Determine the average normal stress at

points D and E and represent the stress on a volume element located at

each of these points

diameter bars of the truss Set P = 40kN.

Problem: Determine the average shear stress developed in pin A of the truss A horizontal force of P = 40kN is applied to joint C Each pin has a

diameter of 25 mm and is subjected to double shear

Determine the average normal stress in each member due to the loading.

120cm

160cm 160cm

suspended from cables AB and AC, determine the average normal stress developed in the cables The diameters of AB and AC are 12 mm and 10

mm, respectively

Q Q

ave



ave

Q A

the equations of equilibrium

A –area at the section

2 2

4 4

P d







Double Shear

2

2 2

2 2

2 2

The factor of safety (F.S)

Here: F fail − the failure load (found from experimental testing of the material)

F allow − the allowable load

We have:

z

P A

Design of simple connection

For normal force requirement

allow

P A





allow

Q A

will support if the average shear stress in each pin is not to allowed toexceed 60 MPa All pins are subjected to double shear as shown, andeach has a diameter of 18mm

= 15kN, determine the average shear stress developed in the pins at A, B, and C All pins are in double shear as shown, and each has a diameter of

18 mm

travel along the bottom flange of the beam, 0,3m ≤ x ≤ 3,6m If the hoist

is rated to support a maximum of 7000 N, determine the maximum

average normal stress in the 18mm-diameter tie rod BC and the maximum average shear stress in the 15mm-diameter pin at B.

the bell crank so that the average normal stress developed in the 10-mm

diameter rod CD, and the average shear stress developed in the 6-mm diameter double sheared pin B not exceed 175MPa and 75MPa,

respectively

plate A Determine the average shear stress in the plate due to this

loading

creates in the metal plate through section AC and BD Also, what is the

bearing stress developed on the surface of the plate under the shaft?

5 kN Determine the average normal stress in each rod and the average

shear stress in the pin A between the members

5.2.2 Shear Strain

2



The approximate angles between sides, again originally defined by the

sides Δx, Δy and Δz are

1  xx 1  yy 1  zz

xz yz

2

The approximate lengths of the sides of the parallelepiped are

Notice that the normal strains cause a change in volume of rectangular element, whereas the shear strain cause a change in shape

shown in fig a Determine (a) the average normal strain along the side

AB, and (b) the average shear strain in the plate at A relative to the and y

axes

Part (a)

• Line AB, coincident with the y axis, becomes line after deformation,

thus the length of this line is

• The average normal strain for AB is therefore

• The negative sign indicates the strain causes a contraction of AB.

250 2 3 248 018 mm '   2  2 

AB

  7 93 10 mm/mm (Ans)

240

250 018

248

avg AB



CE If the load P on the beam causes the end C to be displaced 10 mm

downward, determine the normal strain developed in wires CE and BD.

lever to rotate clockwise about the pin B through an angle of 20.Determine the average normal strain developed in each wire The wiresare unstretched when the lever is in the horizontal position.

causes point A to be displaced horizontally 2 mm, determine the normal

strain developed in each wire

shear strain γ xy at corners A and B if the plastic distorts as shown by the

dashed lines

Problem: The piece of plastic is originally rectangular Determine theshear strain γ xy at corners D and C if the plastic distorts as shown by the

dashed lines