WORLD FOR CIVIL

Topic 5.2: Beams - Bending Stress (cont)

We continue our discussion of the Flexure Formula., M y / I, shown in its general form. In Bending Stress Example 1, we applied the flexure formula to a loaded rectangular beam. Rectangular metal beams are not normally used in large building construction, as they are not very efficient, rather metal I beams, and T beams are used instead. Rectangular beams are used in wood construction where the cost of using a metal I or T beam is excessive, and the lower cost of wood rectangular beams offsets their lower efficiency.

I-Beams
We will now consider an I-beam example. In Diagram 1, we have shown an I-beam cross section. The top and bottom sections are known was the flanges, and the connecting region is known as the web. The horizontal line passing through the center (centroid) of the beam is the neutral axis.

In the sample Beam Data Table shown below, beam information is shown. The Beam Designation, such as W 8 x 20, gives us the following information. The "W" indicates a Wide Flange beam. The "8" gives the approximate depth of the beam in inches. (Notice the actual depth of the beam is 8.14"). The "20" indicates the weight per foot of the beam in a standard type of steel. Thus the first beam in the table, W 8 x 20, is a wide flange, eight inch deep I-beam, with a weight of 20 lb/ft. Additional information given in the table includes the flange width and depth, the web thickness, the moment of inertia, I, about the x-x neutral axis of the beam (shown in Diagram 1), something called the section modulus, S, which we will discuss shortly, and the radius of gyration, r, with respect to the x-x neutral axis. Additionally, there is also a moment of inertia, section modulus, and radius of gyration about the y-y neutral axis. (This would apply if we flipped the beam on its side and loaded it in that orientation. This is not usually done and so we will not expect to use the y-y axis information.).(You may wish to print out this topic and the beam tables to study them more effectively.)

I-Beams	-	-	Flange	Flange	Web	Cross	Section	Info.	Cross	Section	Info.
Designation	Area	Depth	Width	thick	thick	x-x axis	x-x axis	x-x axis	y-y axis	y-y axis	y-y axis
-	A	d	bf	tf	tw	I	S	r	I	S	r
-	in2	in	in	in	in	in4	in3	in	in4	in3	in
W 8x20	5.89	8.14	5.268	0.378	0.248	69.4	17.0	3.43	9.22	3.50	1.25
W 8x17	5.01	8.00	5.250	0.308	0.230	56.6	14.1	3.36	7.44	2.83	1.22
W 10x45	13.20	10.12	8.022	0.618	0.350	249.0	49.1	4.33	53.20	13.30	2.00
W 10x39	11.50	9.94	7.990	0.528	0.318	210.0	42.2	4.27	44.90	11.20	1.98
W 10x33	9.71	9.75	7.964	0.433	0.292	171.0	35.0	4.20	36.50	9.16	1.94
W 12x22	6.47	12.31	4.030	0.424	0.260	156.0	25.3	4.91	4.64	2.31	0.85
W 12x19	5.59	12.16	4.007	0.349	0.237	130.0	21.3	4.82	3.76	1.88	0.82
W 12x31	9.13	12.09	6.525	0.465	0.265	239.0	39.5	5.12	21.60	6.61	1.54
W 14x38	11.20	14.12	6.776	0.513	0.313	386.0	54.7	5.88	26.60	7.86	1.54
W 14x34	10.00	14.00	6.750	0.453	0.287	340.0	48.6	5.83	23.30	6.89	1.52
W 16x50	14.70	16.25	7.073	0.628	0.380	657.0	80.8	6.68	37.10	10.50	1.59
W 16x40	11.80	16.00	7.000	0.503	0.307	517.0	64.6	6.62	28.80	8.23	1.56
W 16x36	10.60	15.85	6.992	0.428	0.299	447.0	56.5	6.50	24.40	6.99	1.52
W 24x94	27.70	24.29	9.061	0.872	0.516	2690.0	221.0	9.86	108.00	23.90	1.98
W 24x76	22.40	23.91	8.985	0.682	0.400	2100.0	176.0	9.69	82.60	18.40	1.92
W 24x68	20.00	23.71	8.961	0.582	0.416	1820.0	153.0	9.53	70.00	15.60	1.87

We now look at an example of a loaded I-beam. Please select Topic 5.2a: Bending Stress - Example 2

T-Beams

We will next consider a T-beam example. In Diagram 2, we have shown a T-beam cross section. The top horizontal section is known as the flange, and the vertical section is known as the stem. The horizontal line passing through a portion of the T is the neutral axis. In the diagram, y is the distance from the top of the T to the neutral axis of the beam.

In the Beam Data Table shown below, beam information is given. The Beam Designation, such as WT 6 x 11, gives us the following information. The "WT" indicates a Wide Flange Tee beam. The "6" gives the approximate depth of the beam in inches. (Notice the actual depth of the beam is 6.16"). The "11" indicates the weight per foot of the beam in a standard type of steel. Thus the first beam in the table, W 6 x 11, is a wide flange, six inch deep T-beam, with a weight of 11 lb./ft. Additional information given in the table includes the flange width and depth, the stem thickness, the moment of inertia, I, about the x-x neutral axis of the beam (shown in Diagram 2), the section modulus, S, the radius of gyration, r, with respect to the x-x neutral axis, and y, the distance from the top of the tee to the neutral axis of the beam.

T-Beams	-	Depth	Flange	Flange	Stem	-	Cross	Section	Info.	-
Designation	Area	of T	Width	thick	thick	-	x-x axis	x-x axis	x-x axis	x-x axis
-	A	d	bf	tf	tw	d/tw	I	S	r	y
-	in2	in	in	in	in	-	in4	in3	in	in
WT 6x11	3.24	6.16	4.030	0.424	0.260	23.70	11.70	2.590	1.900	1.630
WT 6x9.5	2.80	6.08	4.007	0.349	0.237	25.70	10.20	2.300	1.910	1.650
WT 7x26.5	7.79	6.97	8.062	0.658	0.370	18.80	27.70	4.960	1.880	1.380
WT 7 x24	7.06	6.91	8.031	0.593	0.339	20.40	24.90	4.490	1.880	1.350
WT 8x25	7.36	8.13	7.073	0.628	0.380	21.40	42.20	6.770	2.400	1.890
WT 8x20	5.89	8.00	7.000	0.503	0.307	26.10	33.20	5.380	2.370	1.820
WT 8x18	5.30	7.93	6.992	0.428	0.299	26.50	30.80	5.110	2.410	1.890
WT10.5 x34	10.00	10.57	8.270	0.685	0.430	24.60	103.00	12.900	3.200	2.590
WT 10x31	9.13	10.50	8.240	0.615	0.400	26.20	93.80	11.900	3.210	2.580
WT 12x47	13.80	12.15	9.061	0.872	0.516	23.50	186.00	20.300	3.670	3.000
WT 12x42	12.40	12.05	9.015	0.772	0.470	25.60	166.00	18.300	3.660	2.970
WT 12x38	11.20	11.96	8.985	0.682	0.440	27.20	151.00	16.900	3.680	2.990
WT 15x66	19.40	15.15	10.551	1.000	0.615	24.60	421.00	37.400	4.650	3.900
WT 15x58	17.10	15.00	10.500	0.850	0.564	26.60	372.00	33.600	4.670	3.930
WT 15x54	15.90	14.91	10.484	0.760	0.548	27.20	350.00	32.100	4.690	4.020
WT18x97	28.60	18.24	12.117	1.260	0.770	23.70	905.00	67.400	5.630	4.810
WT 18x75	22.10	17.92	11.972	0.940	0.625	28.70	698.00	53.100	5.620	4.780

We now look at an example of a loaded T-beam. Please Select: Topic 5.2b: Bending Stress - Example 3