ES 13 1st Exam Summary

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MECHANICS OF DEFORMABLE BODIES I INTRODUCTION TO MECHANICS OF DEFORMABLE BODIES

1 Exam Coverage REACTIONS and INTERNAL FORCES Reactions - surface forces that develop at the supports or points of contact between bodies

LOAD CLASSIFICATION A. According to Time

T 1. Static Load - load that is gradually applied for which equilibrium is achieved at a very short time Cable

2. Sustained Load - load that is constant over a long period of time 3. Impact Load - load that is applied in a rapid and impulsive manner 4. Repeated Load - load that is applied and removed successively

R

Roller

B. According to Distribution Rx 1. Concentrated Load - point load

Ry

Hinge

2. Distributed Load - a load distributed along a line or a surface

Distributed Load

General Force System  Fx  0

 Fy  0  Fz  0



Ry

 Mx  0  My  0  Mz  0

Coplanar Force System  Fx  0

 Fy  0  Mp  0

1. Axial Load – load that is applied along the axis of the member 2. Torsional Load - load that twists a member  3. Flexural/Bending Load - load that is applied transversely to the longitudinal axis of the member

M

Equations of Equilibrium 

C. According to Location and Method of Application

Rx

Fixed

Concentrated Load

Concurrent Force System  Fx  0

 Fy  0

4. Combined Loading - any combination of the first 3 above Internal Forces - forces developed within the body of a member due to application of external loads

INSTITUTE OF CIVIL ENGINEERING

From the lectures of Jaime Hernandez,Jr., Ian Sison, Glenn Pintor, Juan Michael Sargado, Romeo Longalong, and Raniel Suiza

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MECHANICS OF DEFORMABLE BODIES I

1 Exam Coverage

STRESS CONCEPTS

Note:  

Stress - intensity of load / force per unit area (P/A) Units: MPa, kPa, Pa, psi, ksi

The distribution of normal stresses is statically indeterminate. In order to assume a uniform distribution of stresses, the line of action of the concentrated load should pass through the centroid of the section (centric loading).

Normal Stress,  = N/A - stress acting perpendicular to the surface of a cross section. Shear Stress,  = V/A - stress acting parallel or tangent to the surface of a cross section. a

 Area, A

N



V a

NORMAL STRESS Axial Stress – stress resulting from axial loading.

  lim

A0

F A

 ave 

Bearing Stress - for connections, it is the compressive stress developed in the members it connects.

P A

P   ave A   dF    dA A

b 


P P  A td


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MECHANICS OF DEFORMABLE BODIES I SHEAR STRESS

1 Exam Coverage STRAIN CONCEPTS

Simple Shear Stress - stress resulting from transverse loading.

DEFORMATION - change in the shape and size of a body subjected to an external force or a temperature change.

In connections (e.g. pins, bolts, and rivets) DISPLACEMENT - measures the movement of a point or a particle in a body. Single Shear STRAIN - describes the deformation of a body. Normal Strain - elongation or contraction of an element per unit of length. Unit: dimensionless (but usually expressed also as mm/m or m/m)

Double Shear

 avg 

L Li

where

L  algebraic change in member length, meters Li  initial length of member, meters

Punching Stress – stress resulting from pressing or punching.

Shear Strain - angular change between two perpendicular line segments. Unit: radians





  ' where 2   shear strain

 ' measured angle between two lines initially (mutually) perpendicular to each other



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MECHANICS OF DEFORMABLE BODIES I MATERIAL PROPERTIES

1 Exam Coverage ELASTIC BEHAVIOR

NORMAL STRESS-STRAIN DIAGRAM: TENSILE TEST

Proportional Limit - point at which when the specimen is unloaded, it returns to its original length. Elastic Limit - point at which stress is no longer proportional to strain, but still exhibits elastic behavior.

application of incremental loads... P 1  1 Ainitial

2 

P1  P P  n  P ...  n  1 Ainitial Ainitial

PLASTIC BEHAVIOR Yield Point - point at which the specimen continues to deform without further increase in load; deformation becomes permanent.

results to further elongation of specimen.

1 

L1 Linitial

2 

L2 Linitial

...  n 

Ln Linitial

Strain Hardening - region after the end of yielding where additional loads can be applied until the ultimate stress is reached. Necking - region wherein there is reduction in cross-sectional area of the specimen which signifies decrease in load-carrying capacity of the material. CONVENTIONAL and TRUE STRESS-STRAIN DIAGRAM

application of incremental loads...

n 

P1  n  P Ainitial

 n 

P1  n  P Ainstant

results to further elongation of specimen.

n 


Ln Linitial


MECHANICS OF DEFORMABLE BODIES I HOOKE’S LAW for NORMAL STRESS

DUCTILE and BRITTLE MATERIALS

DESIGN PHILOSOPHY -

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1 Exam Coverage

Ductile Materials

Design engineering structures to undergo small deformations that involve only the straight line portion for the stress-strain diagram.

HOOKE’S LAW -

For the initial portion of the stress-strain diagram, stress is proportional to strain. It is defined by the equation

 E where E is called the modulus of elasticity or the Young’s Modulus.

SHEAR STRESS-STRAIN DIAGRAM Brittle Materials

HOOKE’S LAW for SHEAR STRESS -

For the initial portion of the stress-strain diagram, stress is proportional to strain. It is defined by the equation

  G where G is called the shear modulus of elasticity or the modulus of rigidity or the Kirchoff’s Modulus.



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MECHANICS OF DEFORMABLE BODIES I DUCTILITY of a MATERIAL

1 Exam Coverage POISSON EFFECT

STRAIN ENERGY - the energy that a material tends to store internally throughout its volume during deformation.

-

The elongation in one direction is accompanied by a contraction in the other directions.

U  1 / 2 V STRAIN ENERGY DENSITY - strain energy per unit volume; area under the stress-strain diagram.

u

U 1   V 2

Modulus of Resilience, ur - it is the area under the stress-strain diagram where stress is proportional to strain. 2 1 1  pl u r   pl  pl  2 2 E

Modulus of Toughness, ut - indicates the strain-energy density of the material just before it fractures; it is the area under the entire stressstrain diagram.

 long

  L

 lat

'  r

Poisson’s Ratio - ratio of the lateral strain to the longitudinal strain.

 

 lat  long

  long  

 lat 

RELATIONSHIP BETWEEN E, G, and ν

G


E 2 1  


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MECHANICS OF DEFORMABLE BODIES I GENERALIZED HOOKE’S LAW -

1 Exam Coverage ALLOWABLE STRESSES and FACTORS OF SAFETY

For an element subjected to multi-axial loading, the normal strain components resulting from the stress components may be determined from the principle of superposition. This requires that (a) strain is linearly related to stress and (b) deformations are small.

DESIGN PHILOSOPHY -

Design structural members or machine components such that the working stresses (or actual load) are less than the ultimate strength (or ultimate load) of the material.

FACTOR OF SAFETY - measure of safety of a structural or machine element under its design or applied loads.

Factors affecting FS: 

uncertainty in material properties (dimensional tolerances, residual stresses due to uneven cooling) uncertainty of loadings (e.g. change in occupancy) importance of member to integrity of whole structure (bracing/secondary members lower FS than primary FS) risk to life and property uncertainty of analyses (approximate analyses: use larger FS) number of loading cycles (effect of fatigue may result to sudden failure) types of failure (brittle vs. ductile materials – former must adopt larger FS) maintenance requirements and deterioration effects (conditions that expose member to elements require larger FS, e.g. reinforced concrete requirements: soil, elements) influence on machine function

 

x   y   z  

 x  y  z E



 x E



E



 y  z E



 x  y E



E

E



E

z E

     

Application of FS: Design vs. Analysis Design

when designing,

Working Stress - expected stress once the element is in service.

F .S . 

strength of material, R working stress, S

Analysis Actual Stress - resulting stress in the element when it is already in actual use.


actual loading, F .S . 

strength of material, R actual stress, S


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1 Exam Coverage

STRESSES and DEFORMATIONS ARISING from AXIAL LOADING

Varying Load and Cross-Sectional Area If the bar is subjected to several different axial forces, or the cross-sectional area or modulus of elasticity changes abruptly from one region of the bar to the next, the above equation can be applied to each segment of the bar where these quantities are all constant.

AXIAL DEFORMATION FORMULA Consider a bar, which has a cross-sectional area that gradually varies along its length, L x

dx

dx

P1

P2

L



P(x)

P(x)

PL

 AE

Sign Convention: P - positive if tensile P - negative if compressive

 - positive (elongation)  - negative (compression)



The stress and strain in the element are



Procedure for Analysis: a. Obtain the internal axial force P - by method of sections and equations of equilibrium - if P varies along the member's length, determine P(x) - If several constant external forces act on the member, the internal force between any two external forces must then be determined. For convenience, construct a normal-force diagram. b. Compute the displacement / deformation, 

d

P( x ) A( x )



d dx

From Hooke's law,

  E P( x ) d E A( x ) dx



d 

P( x )dx A( x )E

Consider the figure below, the solid lines represent the unstrained (unloaded) configuration of the system and the dashed lines represent the configuration due to a force applied at B.

Integrating, L



P( x )dx A( x )E 0



PRINCIPLE:

A

where  = displacement of one point on the bar relative to another point L = distance between the points P(x) = internal axial force at the section, located a distance x from one end A(x) = cross-sectional area of the bar, expressed as a function of x E = modulus of elasticity

C L R

Constant Load and Cross-Sectional Area



B



PL AE

P

P E, A L


D AB  y BD  x

y 

For very small displacements, the axial deformation in any bar may be assumed equal to the component of the displacement of one end of the bar (relative to the other end) taken in the direction of the unstrained orientation of the bar.

B' x


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1 Exam Coverage

ANALYSIS of STATICALLY DETERMINATE AXIALLY LOADED MEMBERS Consider,

TEMPERATURE EFFECTS

PA PB 30

45

A change in temperature can cause a material to change its dimensions. If the temperature increases, generally a material expands, whereas if the temperature decreases, the material will contract. The deformation due to a temperature change of a material is given by,

30

45

 T  LT

B A

100 kN

where

Equations of Equilibrium (CFS) = 2 Fx = 0 Fy = 0 Number of Unknowns = 2

100 kN

ANALYSIS of STATICALLY INDETERMINATE AXIALLY LOADED MEMBERS Consider,

PA

PC PB

45

30

C

45

30

B A

100 kN 100 kN

Equations of Equilibrium (CFS) = 2 Fx = 0 Fy = 0 Number of Unknowns = 3

3. 4.

Temperature Effects on Statically Determinate Members Members are free to expand or contract when they undergo a temperature change. Thermal stresses are zero. Temperature Effects on Statically Indeterminate Members Thermal displacements on members can be constrained by the supports, producing thermal stresses that must be considered in design. THERMAL STRESS - induced stress when the body that is subjected to a temperature change is restrained (free movement prevented). For elastic action, thermal stresses are computed by: 1. Assuming that the restraining influence has been removed and the member permitted to expand or contract freely 2. Applying forces that cause the member to assume the configuration dictated by the restraining influence. A

Procedures: 1. 2.

 - coefficient of thermal expansion (1/C) L - length of the member T - algebraic change in temperature (C)

Draw a Free Body Diagram (FBD). Recognize the type of force system on the FBD and note the number of independent equations of equilibrium available for the system. If the number of unknowns exceeds the number of equilibrium equations, a deformation (compatibility) equation must be written for each extra unknown. When the number of independent equations and deformation equations equal the number of unknowns, the equations can be solved simultaneously. Deformations and forces must be related in order to solve the equations simultaneously.

B L

A

A

 P  T

T

PL  LT AE   TE

B'

B

P

Thermal Stress

P




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1 Exam Coverage

PRINCIPLE OF SUPERPOSITION This principle states that stresses due to different loads may be computed separately and added algebraically, provided that the sum of the stresses does not exceed the proportional limit of the material and that the structure remains stable. GENERALIZED HOOKE’S LAW For multi-axial loading, temperature effects can be incorporated by adding its strain components to the previously discussed strain components in Hooke’s Law using the principle of superposition.

 x  y

x  

E

y   z  



 x E



E



 z



 z

y E

 x  y E



E

E



E

z E

 T  T  T



ES 13 1st Exam Summary

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