At a point subjected to plane stress, the stresses are σx = –40 MPa, σy = 30 MPa, and τxy = –90 MPa. Determine the angle θp corresponding to the orientation of the principal planes at the point.

A simply supported beam with dimensions of b = 12 in., h = 26 in., d = 23.5 in., and L = 16 ft supports a uniform service (unfactored) dead load consisting of its own weight plus 1.2 kips/ft and a uniform service (unfactored) live load of 0.9 kips/ft. The concrete is normal-weight concrete. Determine the moment due to the factored loads, Mu.      

A classmate has suggested a beam design that has dimensions of b = 14 in., h = 30 in., and d = 27.5 in. and is reinforced with 5 No. 14 bars. Assuming a cover of 1.5 in. and a stirrup diameter of 0.5 in., does the reinforcement fit within a single row as suggested?

Compute φVn for the cross section shown. Assume f’c = 7,000 psi, fyt = 60,000 psi, b1 = 29 in., b2 = 12 in., d1 = 8 in., d2 = 16 in., and that there are three No. 7 longitudinal tension bars and No. 3 stirrups at 8 in. o.c.

If φVc = 58 kip and Vu = 21 kip, are stirrups required?

A beam has dimensions of b = 14 in., h = 30 in., d’ = 2.5 in., and d = 27.5 in. It is reinforced with 2 No. 5 bars on the compression side and 4 No. 8 bars on the tension side. The concrete strength is 2,800 psi, and the yield strength of the reinforcement is 60,000 psi. If the depth of the neutral axis is c = 5.68 in., determine the strain in the compression reinforcement, ε’s.

Isolated beams with unsymmetrical cross sections can deflect vertically and laterally at the same time.

A beam has dimensions of b = 14 in., h = 26 in., d’ = 2.5 in., and d = 23.5 in. It is reinforced with 2 No. 5 bars on the compression side and 4 No. 8 bars on the tension side. The concrete strength is 3,100 psi, and the yield strength of the reinforcement is 60,000 psi. If the depth of the neutral axis is c = 5.205 in., determine the strain in the compression reinforcement, ε’s.

Compute φVn for the cross section shown. Assume f’c = 3,000 psi, fyt = 40,000 psi, d = 24 in., w = 18.5 in., t = 4 in., and that there are four No. 9 longitudinal tension bars and No. 4 stirrups at 8 in. o.c.

A beam has dimensions of b = 14 in., h = 26 in., d’ = 2.5 in., and d = 23.5 in. It is reinforced with 2 No. 5 bars on the compression side and 4 No. 8 bars on the tension side. The concrete strength is 2,800 psi, and the yield strength of the reinforcement is 60,000 psi. For the results given below, determine the strength Mn for this beam.a = 4.828 in. (depth of concrete stress block)Cc = 159.4 kips (compressive force in concrete)Cs = 30.2 kips (compressive force in steel)T = 189.6 kips (tensile force in steel)