INTRODUCTION TO BEAMS

Transcription

1 CHAPTER Structural Steel Design LRFD etho INTRODUCTION TO BEAS Thir Eition A. J. Clark School of Engineering Department of Civil an Environmental Engineering Part II Structural Steel Design an Analsis 8b FALL 00 B Dr. Ibrahim. Assakkaf ENCE 55 - Introuction to Structural Design Department of Civil an Environmental Engineering Universit of arlan, College Park CHAPTER 8b. INTRODUCTION TO BEAS Slie No. 1 Elastic Design For man ears the elastic theor has been the basis for steel structural esign an analses. This theor is base on the iel stress of a steel structural element. However, nowaas, it has been replace with a more rational an realistic theor, the ultimate stress esign that is base on the plastic capacit of a steel structure. 1

2 CHAPTER 8b. INTRODUCTION TO BEAS Slie No. Elastic Design In the elastic theor, the maximum loa that a structure coul support is assume to equal the loa that cause a stress somewhere in the structure equal the iel stress F of the material. The members were esigne so that compute bening stresses for service loas i not excee the iel stress ivie a factor of safet (e.g., 1.5 to ) CHAPTER 8b. INTRODUCTION TO BEAS Slie No. Elastic Design Elastic Versus Ultimate-base Design of Steel Structures m m R n L i φrn γ ili FS i 1 i 1 ASD LRFD Accoring to ASD, one factor of safet (FS) is use that accounts for the entire uncertaint in loas an strength. Accoring to LRFD (probabilit-base), ifferent partial safet factors for the ifferent loa an strength tpes are use.

3 CHAPTER 8b. INTRODUCTION TO BEAS Slie No. Elastic Design Engineering structures have been esigne for man ears b the allowable stress esign (ASD), or elastic esign with satisfactor results. However, engineers have long been aware that uctile members (e.g., steel) o not fail until a great eal of ieling occurs after iel stress is first reache. This mean that such members have greater margin of safet against collapse than the elastic theor woul seem to suggest. CHAPTER 8b. INTRODUCTION TO BEAS Slie No. 5 The Elastic oulus The iel moment equals the iel stress F times the elastic moulus S: F S (1) where I S c I moment of inertia c istance from N.A.to outer fiber of cross section

4 CHAPTER 8b. INTRODUCTION TO BEAS Slie No. 6 The Elastic oulus The elastic moulus for a rectangular section b as shown in Fig. 1 can be compute b using: The flexural formula, or The internal couple metho Figure 1 b CHAPTER 8b. INTRODUCTION TO BEAS Slie No. 7 The Elastic oulus Using the Flexural Formula Rectangular Cross Section: c F I F S I / c S b I, c S 1 Fb FS 6 I c b /1 / b 6

5 CHAPTER 8b. INTRODUCTION TO BEAS Slie No. 8 The Elastic oulus Using the Internal Couple etho: Rectangular Section: Figure b N.A. Fb Force moment arm F F 1 F b b C F 1 F b b T F F b 6 CHAPTER 8b. INTRODUCTION TO BEAS Slie No. 9 The Plastic oulus The resisting moment at full plasticit can be etermine in a similar manner. The result is the so-calle plastic moment p. It is also the nominal moment of the section, n p () n 5

6 CHAPTER 8b. INTRODUCTION TO BEAS Slie No. 10 The Plastic oulus The plastic ( or nominal) moment equals T or C times the lever arm between them as shown. Figure F N.A. C F b T F b F b F b b Fb Force lever arm T C p F b F CHAPTER 8b. INTRODUCTION TO BEAS Slie No. 11 The Plastic oulus The plastic moment is equal to the iel stress F times the plastic moulus Z. From the foregoing expression for a rectangular section, the plastic moulus Z can be seen to equal b /. p b FZ F b Z 6

7 CHAPTER 8b. INTRODUCTION TO BEAS Slie No. 1 The Plastic oulus The shape factor, which is equal p FZ Z FS S Is also equal to b Z Shape Factor 1.5 S b 6 So, for rectangular section, the shape factor equal 1.5. CHAPTER 8b. INTRODUCTION TO BEAS Slie No. 1 The Plastic oulus Shape Factor Definition The shape factor of a member cross section can be efine as the ratio of the plastic moment p to iel moment. The shape factor equals 1.50 for rectangular cross sections an varies from about 1.10 to 1.0 for stanar rollebeam sections 7

8 CHAPTER 8b. INTRODUCTION TO BEAS Slie No. 1 The Plastic oulus Shape Factor The shape factor Z can be compute from the following expressions: P Shape Factor () Or from Shape Factor Z S () CHAPTER 8b. INTRODUCTION TO BEAS Slie No. 15 The Plastic oulus Neutral Axis for Plastic Conition The neutral axis for plastic conition is ifferent than its counterpart for elastic conition. Unless the section is smmetrical, the neutral axis for the plastic conition will not be in the same location as for the elastic conition. The total internal compression must equal the total internal tension. 8

9 CHAPTER 8b. INTRODUCTION TO BEAS Slie No. 16 The Plastic oulus Neutral Axis for Plastic Conition As all fibers are consiere to have the same stress F in the plastic conition, the areas above an below the plastic neutral axis must be equal. This situation oes not hol for unsmmetrical sections in the elastic conition. CHAPTER 8b. INTRODUCTION TO BEAS Slie No. 17 The Plastic oulus Plastic oulus Definitions The plastic moulus Z is efine as the ratio of the plastic moment p to the iel stress F Y. It can also be efine as the first moment of area about the neutral axis when the areas above an below the neutral axis are equal. 9

10 CHAPTER 8b. INTRODUCTION TO BEAS Slie No. 18 The Plastic oulus Example 1 Determine the iel moment, the plastic or nominal moment p ( n ), an the plastic moulus Z for the simpl supporte beam having the cross section shown in Fig. b. Also calculate the shape factor an nominal loa P n acting transversel through the mispan of the beam. Assume that F Y 50 ksi. CHAPTER 8b. INTRODUCTION TO BEAS Slie No. 19 The Plastic oulus Example 1 (cont ) Figure P n 17 in. 1 ft 1 ft (a) 8 in. (b) 10

11 CHAPTER 8b. INTRODUCTION TO BEAS Slie No. 0 The Plastic oulus Example 1 (cont ) Elastic Calculations: A 15 1 C N.A. 17 in. I 9.97 in. x () + 15() 1 + 8() 1 8 in ()( 16.5) + 15()( 1 8.5) + 8()( 1 0.5) in from lower base ( 9.97) 7( 8.97) 15( 7.06) 1( 6.06) 8 1,67.6 in + 8 in. S I c 1, in 9.97 ( ) FY S ft - kip CHAPTER 8b. INTRODUCTION TO BEAS Slie No. 1 N. A A 1 The Plastic oulus Example 1 (cont ) 8 in. Plastic Calculations: N.A. The areas above an below the neutral axis must be equal for plastic analsis 17 in. A A 1 () + ( 15 )() 1 8() 1 + () N N in N N 8 + N N 11

12 CHAPTER 8b. INTRODUCTION TO BEAS Slie No. The Plastic oulus A 1 Example 1 (cont ) N.A. 17 in. 1 Plastic Calculations (cont ): ()( ) ()( 5.5) + 15()( 1.5) + ()( 1 ) 8 in Z 8 p ( 8) 50 n FZ ft - kip n 950 Shape Factor A 8 in. Note, the shape factor can also be calculate Z Shape Factor S from CHAPTER 8b. INTRODUCTION TO BEAS Slie No. The Plastic oulus Example 1 (cont ) In orer to fin the nominal loa P n, we nee to fin an expression that gives the maximum moment on the beam. This maximum moment occurs at mispan of the simpl supporte beam, an is given b P n 1 ft 1 ft L P L / Pn L 1

13 CHAPTER 8b. INTRODUCTION TO BEAS Slie No. The Plastic oulus Example 1 (cont ) P n Therefore, n P P 950 P n L / ( ) ( 950) Pn L 158. kips 1 A A 1 8 in. N.A. 1 ft 1 ft L 17 in. CHAPTER 8b. INTRODUCTION TO BEAS Slie No. 5 Theor of Plastic Analsis The basic theor of plastic analsis is consiere a major change in the istribution of stresses after the stresses at certain points in a structure reach the iel stress F. The plastic theor implies that those parts of the structure that have been stresse to the iel stress F cannot resist aitional stresses. 1

14 CHAPTER 8b. INTRODUCTION TO BEAS Slie No. 6 Theor of Plastic Analsis The instea will iel the amount require to permit the extra loa or stresses to be transferre to other parts of the structure where the stresses are below the iel stress F, an thus in the elastic range an able to resist increase stress. Plasticit can be sai to serve the purpose of equalizing stresses in cases of overloa. CHAPTER 8b. INTRODUCTION TO BEAS Slie No. 7 Theor of Plastic Analsis Iealize Stress-Strain Diagram for Steel The stress-strain iagram is assume to have the iealize shape shown in Fig. 5. The iel stress an the proportional limit are assume to occur at the same point for this steel. Also, the stress-strain iagram is assume to be a perfectl straight line in the plastic range. Beon the plastic range there is a range of strain harening. 1

15 CHAPTER 8b. INTRODUCTION TO BEAS Slie No. 8 Theor of Plastic Analsis Figure 5. Stress-Strain Diagram for Steel F Plasticit Strain harening Unit Stress, f E Slope f ε ( Elasticit) Unit Strain, ε CHAPTER 8b. INTRODUCTION TO BEAS Slie No. 9 Theor of Plastic Analsis Iealize Stress-Strain Diagram for Steel The strain harening range coul theoreticall permit steel members to withstan aitional stress. However, from a practical stanpoint, the stains occurring are so large that the cannot be consiere. Furthermore, inelastic buckling will limit the abilit of a section to evelop a moment greater than p, even if strain harening is significant. 15