The president of the troubled National Bank of Washington, Dale L. Jernberg, was formally dismissed yesterday along with two of his top officers, as Washington's third largest bank began to show signs of strain from federal investigations and internal power struggles.

Two directors also are resigning, and the American Federation of State County and Municipal Employes is withdrawing $26 million from the bank and shifting the account to a rival bank following months of bad publicity surrounding the National Bank of Washington, according to well-informed sources.

The purge of top officers was engineered by the bank's majority stockholder, the United Mine Workers of America, which owns three-fourths of the bank's stock and has controlled the institution since 1949. Jernberg's departure marks the third time in six years that the union has fired the top officer of the bank.

The dismissals were made in the midst of investigations by a federal grand jury and the Securities and Exchange Commission into a variety of questionable and possibly illegal practices, including allegations that directors placed on the bank's board by the mine workers' union forced through large and uncreditworthy loans on which kickbacks may have been paid and undisclosed partnerships arranged.

In a three-paragraph statement issued yesterday by the bank, Jernberg's departure was said to stem "from a mutual understanding between himself and the bank."

No immediate successor was named to replace Jernberg as chief executive officer, the title that brings with it undisputed operating control of the bank. Frederick M. Henschel, an executive vice president who formerly handled the bank's relations with its unionized employes, was named to a post of chief administrative officer.

The bank has been without a chairman since the death last November of Joseph B. Danzansky.

The bank's official statement made no mention of two other dismissals that were ordered by a new oversight committee set up by mine workers president Sam Church Jr., who was named chairman of the oversight committee. A majority of the committee's members are union associates.

Several sources familiar with the committee's work said that it also had dismissed John A. Gommengenger, executive vice president for credit and loan administration at the bank, and the National Bank of Washington's longtime senior vice president and credit chief, John R. Cassidy, who, after Jernberg, was the highest paid officer of the bank at $84,400. (Jernberg was paid more than $149,000 in 1979.)

Cassidy and Gommengenger could not be reached for comment.

Jernberg will be paid salary and benefits through February, 1982, while payments to the other two officers will be terminated in six months. The bank is still paying more than $100,000 a year in salary to former bank president Sanderson, who was fired by Church last year.

Together, the salaries paid to dismissed officers add up to a significant percentage of the bank's annual earnings, which plummeted 69 percent during the second quarter this year.

The dismissal of the top officers is part of a two-year power struggle that has enveloped the bank. Jernberg's departure is tied not only to an internal investigative report which accused him of overzealously pushing loans to people whom he "perceived" to have influence with union officials, but also to his own personal power struggle with former UMW legal strategist Ronald G. Nathan, who became the bank's general counsel in 1978.

Nathan resigned from the bank last month, and is currently the chief target of the federal grand jury investigation into possible illegal loan practices at the bank.

Jernberg touched off that investigation last April by turning over to federal authorities information showing that Nathan held an undisclosed partnership in a gas well investment venture that had received $1 million in loans from the bank.

Church, sources say, was angered that Jernberg moved against Nathan without consulting with or informing Church.

The reasons for the other dismissals were less clear. Both Cassidy and Gommengenger were involved in processing loans that now are the subjects of investigation. However, sources familiar with the bank's operations during this time say that neither man had sufficient authority to independently control lending decisions of great magnitude.

In the bank's hierarchy, Cassidy answered directly to Gommengenger and Gommengenger answered directly to Jernberg.

The two resigning members of the bank's board of directors are John W. Lyon, 54 and Irving B. Yoskowitz, 34.

Lyon is general manager of Parking Management, Inc., one of the city's largest parking firms, and president of Excavation Construction Co. of Bladensburg. That company is heavily in debt to the bank and last year sought protection from its creditors under federal bankruptcy laws.

Moreover, the bank's internal investigative report, completed last month, concluded that multimillion dollar loans to Lyon's construction firm were given special treatment. When Lyon's company fell into financial trouble, "the bank's lending decisions were unduly lenient, including decisions as to loan security and repayment schedules," according to the report.

The report found no wrongdoing on Lyon's part, but criticized the bank's managers for being overly indulgent to Lyon because of his longtime positions as chairman of the bank's salary and nominating committee, the panel that passes on raises, bonuses and other benefits for officers of the bank.

Yoskowitz, former IBM regional counsel and now deputy general counsel to United Technologies Corp., was nominated to the bank's board in 1978 by Nathan. He has told the bank's board that he will not stand for reelection at the end of the year.

These directors' departures follow the resignations last year of Washington businessmen Foster Shannon and A. J. Somerville Jr. Shannon, partner in the giant real estate firm, Shannon & Luchs, told close friends that he "bailed out" from the bank's board over what he thought were imprudent loan practices and Nathan's overbearing dominance of the bank's executive committee, of which both were members.

Somerville, a fourth-generation Washington businessman and president of the plumbing supply firm that bears his family name, expressed similar concerns after he resigned.

One of the strongest psychological blows to the bank in recent weeks was the decision by the American Federation of State, County and Municipal Employees to pull its $26 million account out of the bank's trust department, according to several sources.

William Lucy, the 46-year-old AFSCME representative on the bank's board could not be reached for comment.But one source at the bank said the decision to phase out the trust account, on which the National Bank of Washington earns several hundred thousand dollars a year in fees, was based solely on negative publicity associated with the federal investigations. Lucy's own position on the board of the bank is said by sources to be extremely tenuous in the face of the loss of the union account.

Meanwhile, this week's turmoil at the bank is expected to charge the atmosphere of next week's board meeting. Henschel begins his stewardship of the institution at what may be the nadir of the bank's public image in a decade.

The bank last week received its annual examination report from the Office of the Comptroller of the Currency, the federal agency that regulates all national banks. One insider at the bank characterized the agency's report and its conclusions as "very bad."