Sample Problem: Find the maximum tensile and compressive stress in the T-beamshown below.80mmg = 10 kN/mZf20mml1下yNX120 mm2.2 m mX: Solution:20mm1. Centroid (neutral surface, neutral axis):y1ZA,y,80×20×10+20×120×(60+20)52mmyZA80×20+20×1202. Moment of inertia:By the Parallel Axis Theorem: I., = I, + Ad?80×203+20×1203+80×20×(52- 10)2 ++20×120×(80-52)21212=764x104 mm4 =7.64×10-6 m416
• Find the maximum tensile and compressive stress in the T-beam shown below. 2.2 m 1 m q = 10 kN/m 20 mm 20 mm 80 mm 120 mm y1 z z1 y 1. Centroid (neutral surface, neutral axis): 80 20 10 20 120 60+20 52 mm 80 20 20 120 i i i A y y A By the Parallel Axis Theorem: 4 4 6 4 2 3 2 3 z 764 1 0 m m 7.64 1 0 m 2 0 120 8 0 5 2 1 2 2 0 120 8 0 2 0 5 2 1 0 1 2 8 0 2 0 I ( ) ( ) 2 z z zz ' ' I I Ad 2. Moment of inertia: Sample Problem • Solution: 16
3. Reaction forces and diagram of1Fbending momentsHx0=MA, → FB=23.27KN5kNm= F =10×3.2- F= 8.73KNO!Xp=0.87m4DBcM(x)= Fx- qx/?x e[0,2.2 m)3.8 kNmM (x)= Ffx + Fs (x- 2.2) - qx/xe[2.2 m,3.2 m]4. Maximum normal stress (At cross-section B)(-5×10°)×(-52×10°) = 34 ×10° Pa = 34 MPa+Tmax7.64 ×10~6(-5×10°)×[(140 -52)×10-3]1 = -57.6×10 Pa = -57.6 MPaOmax7.64 ×10-617
A D B C 3.8 kNm 5 kNm - + xD = 0.87 m 4. Maximum normal stress (At cross-section B) 3 -3 6 max -6 3 3 6 max 6 ( 5 10 ) ( 52 10 ) 34 10 Pa 34 MPa 7.64 10 ( 5 10 ) [(140 52) 10 ] 57.6 10 Pa 57.6 MPa 7.64 10 0 23.27KN M F Ay B FA FB x 3. Reaction forces and diagram of bending moments F F A B 10 3.2 8.73KN 2 2 0,2.2 m 2 2.2 2.2 m,3.2 m 2 A A B qx M x F x x qx M x F x F x x 17
5. Maximum normal stress (At cross-section D)(3.8×103)×[(140- 52)×10-3143.8×10Pa=43.8MPa9nax7.64 ×10-6(3.8 ×103)×[-52 ×10-3]5 kNm7.64×10-6o:Xp=0.87mDBC?3.8 kNm6. Maximum normal stress: Maximum tensile stress: lower edge of cross-section D (43.8 MPa): Maximum compressive stress: lower edge of cross-section B (-57.6MPa).18
A D B C 3.8 kNm 5 kNm - + xD = 0.87 m 3 3 6 max 6 3 3 max 6 (3.8 10 ) [(140 52) 10 ] 43.8 10 Pa 43.8 MPa 7.64 10 (3.8 10 ) [ 52 10 ] 7.64 10 • Maximum tensile stress: lower edge of cross-section D (43.8 MPa). 18 • Maximum compressive stress: lower edge of cross-section B (-57.6 MPa). 5. Maximum normal stress (At cross-section D) 6. Maximum normal stress
Normal Stress Strength Condition. For ductile materialsM≤[α]OmaxWmax.For brittlematerialsMMmaxWWmaxmax: The maximum positive and negative bending moments in a beammay occur at the following places: (1) a cross section where aconcentrated load is applied and the shear force changes sign, (2) across section where the shear force equals zero, (3) a point of supportwhere a vertical reaction is present, and (4) a cross section where acouple is applied.19
• For brittle materials • For ductile materials Normal Stress Strength Condition max z max M W max max max max , z z M M W W 19 • The maximum positive and negative bending moments in a beam may occur at the following places: (1) a cross section where a concentrated load is applied and the shear force changes sign, (2) a cross section where the shear force equals zero, (3) a point of support where a vertical reaction is present, and (4) a cross section where a couple is applied
Remarks on Strength Condition. The maximum tensile stress and the maximum compressive stresssometimes don't occur on the same cross-section.. Usually, the allowable bending stress is slightly higher than theallowable uniaxial tensile/compressive stress. This is because thebending stress only takes extremities at the upper/lower edges ofbending beams while the maximum axial stress is uniformlydistributed on bar cross-sectionsStrength check: Three types problem that aretypically addressed by strengthCross-sectiondesignanalysis:Allowableload20
• Three types problem that are typically addressed by strength analysis: Strength check Cross-section design Allowable load • The maximum tensile stress and the maximum compressive stress sometimes don’t occur on the same cross-section. Remarks on Strength Condition • Usually, the allowable bending stress is slightly higher than the allowable uniaxial tensile/compressive stress. This is because the bending stress only takes extremities at the upper/lower edges of bending beams while the maximum axial stress is uniformly distributed on bar cross-sections. 20