MATERIALS AND MECHANICS OF BENDING

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1 HAPTER Reinforced oncrete Design Fifth Edition MATERIALS AND MEHANIS OF BENDING A. J. lark School of Engineering Department of ivil and Environmental Engineering Part I oncrete Design and Analysis b FALL 00 By Dr. Ibrahim. Assakkaf ENE 55 - Introduction to Structural Design Department of ivil and Environmental Engineering University of Maryland, ollege Park HAPTER b. MATERIALS AND MEHANIS OF BENDING Slide No. oncrete in Tension oncrete tensile stresses occur as a result of shear, torsion, and other actions, and in most cases member behavior changes upon cracking. It is therefore important to be able to predict, with reasonable accuracy, the tensile strength of concrete. The tensile and compressive strengths of concrete are not proportional, and an increase in compressive strength is accompanied by smaller percentage increase in tensile strength.

2 HAPTER b. MATERIALS AND MEHANIS OF BENDING Slide No. oncrete in Tension The tensile strength of normal-weight concrete in flexure is about 0% to 5% of the compressive strength. There are considerable experimental difficulties in determining the true tensile strength of concrete. The true tensile strength of concrete is difficult to determine. HAPTER b. MATERIALS AND MEHANIS OF BENDING Slide No. oncrete in Tension One common approach is to use the modulus of rupture f r. The modulus of rupture is the maximum tensile bending stress in a plain concrete test beam at failure. Neutral Axis Max. Tensile Stress

3 HAPTER b. MATERIALS AND MEHANIS OF BENDING Slide No. 4 oncrete in Tension AI ode Recommendation For normal-weight concrete, the AI ode recommends that the modulus of rupture f r be taken as f. 5 7 () r f c where f r in psi. HAPTER b. MATERIALS AND MEHANIS OF BENDING Slide No. 5 oncrete in Tension racking Moment, M cr The moment that produces a tensile stress just equal to the modulus of rupture is called cracking moment M cr. The Split-ylinder Test The split-cylinder test has also been used to determine the tensile strength of lightweight aggregate concrete. It has been accepted as a good measure of the true tensile strength.

4 HAPTER b. MATERIALS AND MEHANIS OF BENDING Slide No. 6 oncrete in Tension The Split-ylinder Test (cont d) This test uses a standard 6-in.-diameter, in.-long cylinder placed on its in a testing machine (see Fig. ). A compressive line load is applied uniformly along the length of the cylinder. The compressive load produces a transverse tensile stress, and the cylinder will split in half along the diameter when it tensile strength is reached. HAPTER b. MATERIALS AND MEHANIS OF BENDING Slide No. 7 oncrete in Tension Schematic for Split- ylinder Test Figure 4

5 HAPTER b. MATERIALS AND MEHANIS OF BENDING Slide No. 8 oncrete in Tension Splitting Tensile Strength, f ct The tensile splitting stress can be calculated from the following formula: f ct P πld where f cr splitting tensile strength of concrete (psi) P applied load at splitting (lb) L length of cylinder (in.) D diameter of cylinder (in.) () HAPTER b. MATERIALS AND MEHANIS OF BENDING Slide No. 9 Steel is a high-cost material compared with concrete. It follows that the two materials are best used in combination if the concrete is made to resist the compressive stresses and the steel the tensile stresses. oncrete cannot withstand very much tensile stress without cracking. 5

6 HAPTER b. MATERIALS AND MEHANIS OF BENDING Slide No. 0 Reinforced oncrete Beam Figure d b ompression. Tension d - x x b x N.A. σ F y n A s (a) (b) (c) HAPTER b. MATERIALS AND MEHANIS OF BENDING Slide No. It follows that tensile reinforcement must be embedded in the concrete to overcome the deficiency. Forms of Steel Reinforcement Steel Reinforcing Bars Welded wire fabric composed of steel wire. Structural Steel Shapes Steel Pipes. 6

7 HAPTER b. MATERIALS AND MEHANIS OF BENDING Slide No. HAPTER b. MATERIALS AND MEHANIS OF BENDING Slide No. Reinforcing Bars (rebars) The specifications for steel reinforcement published by the American Society for Testing and Materials (ASTM) are generally accepted for steel used in reinforced concrete construction in the United States and are identified in the AI ode. 7

8 HAPTER b. MATERIALS AND MEHANIS OF BENDING Slide No. 4 Reinforcing Bars (rebars) These bars are readily available in straight length of 60 ft. The bars vary in designation from No. through No. With additional bars: No. 4 and No. 8 HAPTER b. MATERIALS AND MEHANIS OF BENDING Slide No. 5 Table. ASTM Standard - English Reinforcing Bars Bar Designation Diameter in Area in Weight lb/ft # [#0] #4 [#] #5 [#6] #6 [#9] #7 [#] #8 [#5] #9 [#9] #0 [#] # [#6] #4 [#4] #8 [#57] Note: Metric designations are in brackets 8

9 HAPTER b. MATERIALS AND MEHANIS OF BENDING Slide No. 6 Table. ASTM Standard - Metric Reinforcing Bars Bar Designation Diameter mm Area mm Mass kg/m #0 [#] # [#4] #6 [#5] #9 [#6] # [#7] #5 [#8] #9 [#9] # [#0] #6 [#] #4 [#4] #57 [#8] Note: English designations are in brackets HAPTER b. MATERIALS AND MEHANIS OF BENDING Slide No. 7 Yield Stress for Steel Probably the most useful property of reinforced concrete design calculations is the yield stress for steel, f y. A typical stress-strain diagram for reinforcing steel is shown in Fig. a. An idealized stress-strain diagram for reinforcing steel is shown in Fig. b. 9

10 HAPTER b. MATERIALS AND MEHANIS OF BENDING Slide No. 8 Figure Elastic region Elastic region Stress F y Stress F y ε y Strain (a) As Determined by Tensile Test ε y Strain (b) Idealized HAPTER b. MATERIALS AND MEHANIS OF BENDING Slide No. 9 Modulus of Elasticity for Steel The modulus of elasticity for reinforcing steel varies over small range, and has been adopted by the AI ode as E 9,000,000 psi 9,000 ksi 0

11 HAPTER b. MATERIALS AND MEHANIS OF BENDING Slide No. 0 Introduction The most common type of structural member is a beam. In actual structures beams can be found in an infinite variety of Sizes Shapes, and Orientations HAPTER b. MATERIALS AND MEHANIS OF BENDING Slide No. Introduction Definition A beam may be defined as a member whose length is relatively large in comparison with its thickness and depth, and which is loaded with transverse loads that produce significant bending effects as oppose to twisting or axial effects

12 HAPTER b. MATERIALS AND MEHANIS OF BENDING Slide No. Pure Bending: Prismatic members subjected to equal and opposite couples acting in the same longitudinal plane HAPTER b. MATERIALS AND MEHANIS OF BENDING Slide No. Flexural Normal Stress For flexural loading and linearly elastic action, the neutral axis passes through the centroid of the cross section of the beam

13 HAPTER b. MATERIALS AND MEHANIS OF BENDING Slide No. 4 The elastic flexural formula for normal stress is given by Mc f b () I where f b calculated bending stress at outer fiber of the cross section M the applied moment c distance from the neutral axis to the outside tension or compression fiber of the beam I moment of inertia of the cross section about neutral axis HAPTER b. MATERIALS AND MEHANIS OF BENDING Slide No. 5 By rearranging the flexure formula, the maximum moment that may be applied to the beam cross section, called the resisting moment, M R, is given by M Fb I c R (4) Where F b the allowable bending stress

14 HAPTER b. MATERIALS AND MEHANIS OF BENDING Slide No. 6 Example Determine the maximum flexural stress produced by a resisting moment M of ft-lb if the beam has the cross section shown in the figure. 6 6 y HAPTER b. MATERIALS AND MEHANIS OF BENDING Slide No. 7 x Example (cont d) 6 First, we need to locate the neutral axis from the om edge: 5 y y ten ()( 6) + ( + )( 6) 6 y com Max.Stress f Mc I b y c max 4

15 y HAPTER b. MATERIALS AND MEHANIS OF BENDING Slide No. 8 x Example (cont d) 6 Find the moment of inertia I with respect to the x axis using parallel axis-theorem: 5 ( ) ( 6) 6 I + ( 6 )( ) in ( 6)( ) () 5 (5 ) Max. Stress (com).ksi 6 HAPTER b. MATERIALS AND MEHANIS OF BENDING Slide No. 9 Internal ouple Method (cont d) The procedure of the flexure formula is easy and straightforward for a beam of known cross section for which the moment of inertia I can be found. However, for a reinforced concrete beam, the use of the flexure formula can be somewhat complicated. The beam in this case is not homogeneous and concrete does not behave elastically. 5

16 HAPTER b. MATERIALS AND MEHANIS OF BENDING Slide No. 0 Internal ouple Method (cont d) In this method, the couple represents an internal resisting moment and is composed of a compressive force and a parallel internal tensile force T as shown in Fig. 4. These two parallel forces and T are separated by a distance Z, called the the moment arm. (Fig. 4) Because that all forces are in equilibrium, therefore, must equal T. HAPTER b. MATERIALS AND MEHANIS OF BENDING Slide No. Internal ouple Method (cont d) y w P Neutral axis entroidal axis Z c x T da y c c y dy R Figure 4 6

17 HAPTER b. MATERIALS AND MEHANIS OF BENDING Slide No. Internal ouple Method (cont d) The internal couple method of determining beam stresses is more general than the flexure formula because it can be applied to homogeneous or non-homogeneous beams having linear or nonlinear stress distributions. For reinforced concrete beam, it has the advantage of using the basic resistance pattern that is found in a beam. y HAPTER b. MATERIALS AND MEHANIS OF BENDING Slide No. x Example 6 Repeat Example using the internal couple method. 5 N.A Z T 7

18 HAPTER b. MATERIALS AND MEHANIS OF BENDING Slide No. 4 Example (cont d) Because of the irregular area for the tension zone, the tensile force T will be broken up into components T, T, and T. Likewise, the moment arm distance Z will be broken up into components Z, Z, and Z, and calculated for each component tensile force to the compressive force as shown in Fig. 5. HAPTER b. MATERIALS AND MEHANIS OF BENDING Slide No. 5 Example (cont d) f top 5 Z Z Z 6 Figure 5 f T f mid T T 8

19 HAPTER b. MATERIALS AND MEHANIS OF BENDING Slide No. 6 Example (cont d) f T f avg T f avg T f avg avg 6 area f area f area f mid f area top mid [()() 5 ] [()( ) ] [( )( 6) ] f mid 5 f f top mid f mid 5 f 4 f f [( )( 6) ] 6 f 6 fmid f top Z Z Z T f mid T T From similar triangles: f f mid f mid f HAPTER b. MATERIALS AND MEHANIS OF BENDING Slide No. 7 Example (cont d) 6 5 f f top Z Z Z T f mid T T T T + T + T 5 f 5 f top top f f + 4 f + 4 f + 6 f + 6 f 6 f f mid 5 f 5 f top f 9

20 HAPTER b. MATERIALS AND MEHANIS OF BENDING Slide No. 8 Example (cont d) Z Z Z 6 () 5 + () () 5 4 in. 6 + in. 7 () ( ) in. 5 f f top Z Z Z T f mid T T M ext M R 5000( ) ZT + ZT + ZT 60,000 Z T + Z T + Z T HAPTER b. MATERIALS AND MEHANIS OF BENDING Slide No. 9 Example (cont d) ,000 4 f Therefore, f ( 4 f ) + ( 4 f ),.5 psi (Tension) The maximum Stress is compressive stress : f max f top f f + f top Z Z Z T f mid 6 f T T (,.5),05.88 psi. ksi (om) 0

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