## 18 - 20 Rectangular beam in maxima and minima problems

Problem 18
The strength of a rectangular beam is proportional to the breadth and the square of the depth. Find the shape of the largest beam that can be cut from a log of given size.

Problem 19
The stiffness of a rectangular beam is proportional to the breadth and the cube of the depth. Find the shape of the stiffest beam that can be cut from a log of given size.

Problem 20
Compare for strength and stiffness both edgewise and sidewise thrust, two beams of equal length, 2 inches by 8 inches and the other 4 inches by 6 inches (See Problem 18 and Problem 19 above). Which shape is more often used for floor joist? Why?

## Solution to Problem 581 | Design for Flexure and Shear

Problem 581
A laminated beam is composed of five planks, each 6 in. by 2 in., glued together to form a section 6 in. wide by 10 in. high. The allowable shear stress in the glue is 90 psi, the allowable shear stress in the wood is 120 psi, and the allowable flexural stress in the wood is 1200 psi. Determine the maximum uniformly distributed load that can be carried by the beam on a 6-ft simple span.

## Solution to Problem 573 | Horizontal Shearing Stress

Problem 573
The cross-section of a beam is an isosceles triangle with vertex uppermost, of altitude h and base b. If V is the vertical shear, show that the maximum shearing stress is 3V / bh located at the midpoint of the altitude.

Solution 573
$f_v = \dfrac{VQ}{Ib}$

## Solution to Problem 516 | Flexure Formula

Problem 516
A timber beam AB, 6 in wide by 10 in deep and 10 ft long, is supported by a guy wire AC in the position shown in Fig. P-516. The beam carries a load, including its own weight, of 500 lb for each foot of its length. Compute the maximum flexural stress at the middle of the beam.

## Solution to Problem 253 Statically Indeterminate

Problem 253
As shown in Fig. P-253, a rigid beam with negligible weight is pinned at one end and attached to two vertical rods. The beam was initially horizontal before the load W = 50 kips was applied. Find the vertical movement of W.

## Solution to Problem 128 Bearing Stress

Problem 128
A W18 × 86 beam is riveted to a W24 × 117 girder by a connection similar to that in Fig. 1-13. The diameter of the rivets is 7/8 in., and the angles are each L4 × 3-1/2 × 3/8 in.. For each rivet, assume that the allowable stresses are τ = 15 ksi and σb = 32 ksi. Find the allowable load on the connection.

## Problem 352 | Equilibrium of Non-Concurrent Force System

Problem 352
A pulley 4 ft in diameter and supporting a load 200 lb is mounted at B on a horizontal beam as shown in Fig. P-352. The beam is supported by a hinge at A and rollers at C. Neglecting the weight of the beam, determine the reactions at A and C.